Subdiffusive Processes Research Articles

Tempered stable Lévy motion driven by stable subordinator

In this article we propose a new model for financial data description. Combining two independent mechanisms, namely the tempered stable process and inverse stable subordinator, we obtain a new model which captures not only the tempered stable character of the underlying data but also such a property as periods in which the values of an asset stay on the same level. Moreover, we classify our system to the family of subdiffusive processes and investigate its tail behavior. We describe in detail testing and estimation procedures for the proposed model. In the last step we calibrate our model to the real data.

Finite-Time Effects and Ultraweak Ergodicity Breaking in Superdiffusive Dynamics

We study the ergodic properties of superdiffusive, spatiotemporally coupled Lévy walk processes. For trajectories of finite duration, we reveal a distinct scatter of the scaling exponents of the time averaged mean squared displacement δx2 around the ensemble value 3-α (1<α<2) ranging from ballistic motion to subdiffusion, in strong contrast to the behavior of subdiffusive processes. In addition we find a significant dependence of the average of δx2 over an ensemble of trajectories as a function of the finite measurement time. This so-called finite-time amplitude depression and the scatter of the scaling exponent is vital in the quantitative evaluation of superdiffusive processes. Comparing the long time average of the second moment with the ensemble mean squared displacement, these only differ by a constant factor, an ultraweak ergodicity breaking.

Spatio-temporal anomalous diffusion imaging: results in controlled phantoms and in excised human meningiomas

Recently, we measured two anomalous diffusion (AD) parameters: the spatial and the temporal AD indices, called γ and α, respectively, by using spectroscopic pulse gradient field methods. We showed that γ quantifies pseudo-superdiffusion processes, while α quantifies subdiffusion processes. Here, we propose γ and α maps obtained in a controlled heterogeneous phantom, comprised of packed micro-beads in water and in excised human meningiomas. In few words, α maps represent the multi-scale spatial distribution of the disorder degree in the system, while γ maps are influenced by local internal gradients, thus highlighting the interface between compartments characterized by different magnetic susceptibility. γ maps were already obtained by means of AD stretched exponential imaging and α-type maps have been recently achieved for fixed rat brain with the aim of highlighting the fractal dimension of specific brain regions. However, to our knowledge, the maps representative of the spatial distribution of α and γ obtained on the same controlled sample and in the same excised tissue have never been compared. Moreover, we show here, for the first time, that α maps are representative of the spatial distribution of the disorder degree of the system.In a first phase, γ and α maps of controlled phantom characterized by an ordered and a disordered rearrangement of packed micro-beads of different sizes in water and by different magnetic susceptibility (Δχ) between beads and water were obtained. In a second phase, we investigated excised human meningiomas of different consistency.Results reported here, obtained at 9.4T, show that α and γ maps are characterized by a different image contrast. Indeed, unlike γ maps, α maps are insensible to (Δχ) and they are sensible to the disorder degree of the microstructural rearrangement. These observations strongly suggest that AD indices α and γ reflect some additional microstructural information which cannot be obtained using conventional diffusion methods based on Gaussian diffusion. Moreover, α and γ maps obtained in excised meningiomas seem to provide more microstructural details above those obtained with conventional DTI analysis, which could be used to improve the classification of meningiomas based on their consistency.

Analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations on a finite domain

Generalized fractional partial differential equations have now found wide application for describing important physical phenomena, such as subdiffusive and superdiffusive processes. However, studies of generalized multi-term time and space fractional partial differential equations are still under development. In this paper, the multi-term time–space Caputo–Riesz fractional advection diffusion equations (MT-TSCR-FADE) with Dirichlet nonhomogeneous boundary conditions are considered. The multi-term time-fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2] and [0,2], respectively. These are called respectively the multi-term time-fractional diffusion terms, the multi-term time-fractional wave terms and the multi-term time-fractional mixed diffusion-wave terms. The space fractional derivatives are defined as Riesz fractional derivatives. Analytical solutions of three types of the MT-TSCR-FADE are derived with Dirichlet boundary conditions. By using Luchkoʼs Theorem (Acta Math. Vietnam., 1999), we proposed some new techniques, such as a spectral representation of the fractional Laplacian operator and the equivalent relationship between fractional Laplacian operator and Riesz fractional derivative, that enabled the derivation of the analytical solutions for the multi-term time–space Caputo–Riesz fractional advection–diffusion equations.

Clathrin-Mediated Endocytosis Introduces a Nonergodic Diffusion Process in the Plasma Membrane

Tracking individual potassium channels in the plasma membrane reveals complex dynamics involving anomalous diffusion. Theoretical models show that anomalous subdiffusion can be caused by several different processes. In particular, transient binding events, modeled by a continuous time random walk (CTRW), may not only induce anomalous subdiffusion but also weak ergodicity breaking, that is, the ensemble and time averages do not coincide. We studied the physical mechanism underlying Kv2.1 and Kv1.4 potassium channel anomalous dynamics by performing time series analysis of extensive single molecule tracking in the membrane of live mammalian cells. We find ample evidence showing that the ensemble and temporal distributions are different. Our data reveal that two anomalous subdiffusion processes simultaneously coexist and only one of them is ergodic. Weak ergodicity breaking is found to be maintained by immobilization events as long as 60 seconds. In the presence of either actin or chlathrin inhibitors, ergodicity is recovered. In order to elucidate the effects of clathrin endocytosis on Kv2.1 trafficking and diffusion, we have performed simultaneous total internal reflection fluorescence imaging of quantum-dot-tagged Kv2.1 and RFP-tagged clathrin. Our results show that Kv2.1 channels are frequently recycled from and to the plasma membrane. Retrieval of Kv2.1 from the membrane is found to be attained via a clathrin-mediated endocytic pathway. These endocytic/insertion processes are needed to maintain a nonergodic CTRW. Furthermore, the Kv2.1 stalling events colocalize with clathrin clusters. These results suggest that abortive endocytic events are responsible for the observed channel immobilization in the plasma membrane.

Numerical analysis and physical simulations for the time fractional radial diffusion equation

We do the numerical analysis and simulations for the time fractional radial diffusion equation used to describe the anomalous subdiffusive transport processes on the symmetric diffusive field. Based on rewriting the equation in a new form, we first present two kinds of implicit finite difference schemes for numerically solving the equation. Then we strictly establish the stability and convergence results. We prove that the two schemes are both unconditionally stable and second order convergent with respect to the maximum norm. Some numerical results are presented to confirm the rates of convergence and the robustness of the numerical schemes. Finally, we do the physical simulations. Some interesting physical phenomena are revealed; we verify that the long time asymptotic survival probability ∝ t − α , but independent of the dimension, where α is the anomalous diffusion exponent.

Path Properties of Subdiffusion—A Martingale Approach

In statistical physics, subdiffusion processes constitute one of the most relevant subclasses of the family of anomalous diffusion models. These processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean-squared displacement. In this article we study sample path properties of subdiffusion. We propose a martingale approach to the stochastic analysis of subdiffusion models. We verify the martingale property, Hölder continuity of the trajectories, and derive the law of large numbers. The precise asymptotic behavior of subdiffusion is obtained in the law of the iterated logarithm. The presented results may be applied to identify the type of subdiffusive dynamics in experimental data.

Quantitative Analysis of Single Particle Trajectories: Mean Maximal Excursion Method

An increasing number of experimental studies employ single particle tracking to probe the physical environment in complex systems. We here propose and discuss what we believe are new methods to analyze the time series of the particle traces, in particular, for subdiffusion phenomena. We discuss the statistical properties of mean maximal excursions (MMEs), i.e., the maximal distance covered by a test particle up to time t. Compared to traditional methods focusing on the mean-squared displacement we show that the MME analysis performs better in the determination of the anomalous diffusion exponent. We also demonstrate that combination of regular moments with moments of the MME method provides additional criteria to determine the exact physical nature of the underlying stochastic subdiffusion processes. We put the methods to test using experimental data as well as simulated time series from different models for normal and anomalous dynamics such as diffusion on fractals, continuous time random walks, and fractional Brownian motion.

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.

Subdiffusion of mixed origins: When ergodicity and nonergodicity coexist

Single particle trajectories are investigated assuming the coexistence of two subdiffusive processes: diffusion on a fractal structure modeling spatial constraints on motion and heavy-tailed continuous time random walks representing energetic or chemical traps. The particles' mean squared displacement is found to depend on the way the mean is taken: temporal averaging over single-particle trajectories differs from averaging over an ensemble of particles. This is shown to stem from subordinating an ergodic anomalous process to a nonergodic one. The result is easily generalized to the subordination of any other ergodic process (i.e., fractional Brownian motion) to a nonergodic one. For certain parameters the ergodic diffusion on the underlying fractal structure dominates the transport yet displaying ergodicity breaking and aging.

Stochastic representation of subdiffusion processes with time-dependent drift

In statistical physics, subdiffusion processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean square displacement. For the mathematical description of subdiffusion, one uses fractional Fokker–Planck equations. In this paper we construct a stochastic process, whose probability density function is the solution of the fractional Fokker–Planck equation with time-dependent drift. We propose a strongly and uniformly convergent approximation scheme which allows us to approximate solutions of the fractional Fokker–Planck equation using Monte Carlo methods. The obtained results for moments of stochastic integrals driven by the inverse α -stable subordinator play a crucial role in the proofs, but may be also of independent interest.

Hydrodynamic behavior of 1D subdiffusive exclusion processes with random conductances

Considerasystemofparticlesperformingnearestneighborrandomwalks on the lattice Z under hard-core interaction. The rate for a jump over a given bond is direction-independent and the inverse of the jump rates are i.i.d. random variables belonging to the domain of attraction of an α-stable law, 0 <α <1. This exclusion process models conduction in strongly disordered 1D media. We prove that, when varying over the disorder and for a suitable slowly varying function L, under the super-diffusive time scaling N 1+1/α L(N ), the density profile evolves as the solution of the random equation ∂t ρ = LW ρ, where LW is the generalized second-order differential operator d du d dW in which W is a double-sided α-stable subordinator. This resultfollowsfromaquenchedhydrodynamiclimitinthecasethatthei.i.d.jumprates are replaced by a suitable array {ξN ,x : x ∈ Z} having same distribution and fulfilling an a.s. invariance principle. We also prove a law of large numbers for a tagged particle.

Anomalous diffusion with absorbing boundary

In a very long Gaussian polymer on time scales shorter than the maximal relaxation time, the mean squared distance traveled by a tagged monomer grows as approximately t(1/2) . We analyze such subdiffusive behavior in the presence of one or two absorbing boundaries and demonstrate the differences between this process and the subdiffusion described by the fractional Fokker-Planck equation. In particular, we show that the mean absorption time of diffuser between two absorbing boundaries is finite. Our results restrict the form of the effective dispersion equation that may describe such subdiffusive processes.

First-Passage Time Distributions for Subdiffusion in Confined Geometry

In this Letter, we derive a relationship between the moments of the first-passage time for a random walk and the first-passage time density for subdiffusive processes modeled by continuous-time random walks. In particular, we show that the exact long-time behavior of the density depends only on the mean first-passage time of the corresponding normal diffusive process. In addition, we give explicit evaluations of the first-passage time distribution for general three-dimensional bounded domains. These results are relevant to systems involving anomalous diffusion in confinements.

The role of the Fox–Wright functions in fractional sub-diffusion of distributed order

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin–Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox–Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.

From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit anomalous diffusion. We consider here the case of subdiffusive processes, which correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to known fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Front propagation in hyperbolic fractional reaction-diffusion equations

From the continuous-time random walk scheme and assuming a Lévy waiting time distribution typical of subdiffusive transport processes, we study a hyperbolic reaction-diffusion equation involving time fractional derivatives. The linear speed selection of wave fronts in this equation is analyzed. When the reaction-diffusion dimensionless number and the fractional index satisfy a certain condition, we find fronts exhibiting an unphysical behavior: they travel faster in the subdiffusive than in the diffusive regime.

Average shape of fluctuations for subdiffusive walks.

We study the average shape of fluctuations for subdiffusive processes, i.e., processes with uncorrelated increments but where the waiting time distribution has a broad power-law tail. This shape is obtained analytically by means of a fractional diffusion approach. We find that, in contrast with processes where the waiting time between increments has finite variance, the fluctuation shape is no longer a semicircle: it tends to adopt a tablelike form as the subdiffusive character of the process increases. The theoretical predictions are compared with numerical simulation results.

Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids.

The problem of the nonlinear dielectric response due to the application of a strong electric field is reconsidered in the context of fractional kinetic equations. To accomplish that, we start from a fractional noninertial Fokker-Planck equation and restrict ourselves to the case of anomalous subdiffusive processes characterized by the critical exponent alpha ranging from 0 to 1, the limit of normal diffusion. In particular, we evaluate the first- and third-order nonlinear harmonic components of the electric polarization in the case of either a pure ac field or a strong dc bias field superimposed on a weak ac field. The stationary regime is therefore calculated from an infinite set of differential recurrence relations by using a perturbation method. The results so obtained are illustrated by three-dimensional dispersion and absorption plots in order to show the influence of alpha. Cole-Cole diagrams are also presented, allowing one to see that the arcs become more and more flattened as alpha-->0, and corresponding to a broadening of the absorption peaks as effectively observed in complex liquids. The theoretical model is supported by comparison with experimental data of the third-order nonlinear dielectric permittivity of a ferroelectric liquid crystal.

Lévy decoupled random walks

A diffusion model based on a continuous time random walk scheme with a separable transition probability density is introduced. The probability density for long jumps is proportional to x −1− γ (a Lévy-like probability density). Even when the probability density for the walker position at time t, P(x;t) , has not a finite second moment when 0< γ<2, it is possible to consider alternative estimators for the width of the distribution. It is then found that any reasonable width estimator will exhibit the same long-time behaviour, since in this limit P( x; t) goes to the distribution L γ ( x/ t α ), a Lévy distribution. The scaling property is verified numerically by means of Monte Carlo simulations. We find that if the waiting time density has a finite first moment then α=1/ γ, while for densities with asymptotic behaviour t −1− β with 0< β<1 (“long tail” densities) it is verified that α= β/ γ. This scaling property ensures that any reasonable estimator of the distribution width will grow as t α in the long-time limit. Based on this long-time behaviour we propose a generalized criterion for the classification in superdiffusive and subdiffusive processes, according to the value of α.