Anomalous Diffusion Processes Research Articles

Closed-form multi-dimensional solutions and asymptotic behaviors for subdiffusive processes with crossovers: I. Retarding case

Numerous anomalous diffusion processes are characterized by crossovers of the scaling exponent in the mean squared displacement at some correlations time. The bi-fractional diffusion equation containing two time-fractional derivatives is a versatile mathematical tool describing specifically retarded subdiffusive transport, in which the scaling exponents acquires a smaller value, i.e., the diffusion becomes even slower after the crossover. We here derive closed-form multi-dimensional solutions for this integro-differential equation in n spatial dimensions by generalizing the classical Schneider-Wyss solution of the fractional diffusion equation with a single fractional derivative. In the two-dimensional case we develop a limiting approach based on the solution of the space-time fractional diffusion equation. The probabilistic interpretation in higher dimensions is discussed. The asymptotic long- and short-time behaviors are derived. It is shown that the solution of the bi-fractional diffusion equation can be interpreted in terms of the Fox H-transform of the Gaussian distribution.

Time-squeezing and time-expanding transformations in harmonic force fields.

A variety of real life phenomena exhibit complex time-inhomogeneous nonlinear diffusive motion in the presence of an external harmonic force. In capturing the characteristics of such dynamics, the class of Ornstein-Uhlenbeck processes, with its physical time appropriately modulated, has long been regarded as the most relevant model on the basis of its mean reversion property. In this paper, we contrast two similar, yet definitely different, time-changing mechanisms in harmonic force fields by systematically deriving and presenting a variety of key properties all at once, that is, whether or not and how those time-changing mechanisms affect the characteristics of mean-reverting diffusion through sample path properties, the marginal probability density function, the asymptotic degeneracy of increments, the stationary law, the second-order structure, and the ensemble- and time-averaged mean square displacements. Some of those properties turn out rather counter-intuitive due to, or irrespective of, possible degeneracy of time-changing mechanisms in the long run. In light of those illustrative comparisons, we examine whether such time-changing operations are worth the additional operational cost, relative to physically relevant characteristics induced, and deduce practical implications and precautions from modeling and inference perspectives, for instance, on the experimental setup involving those anomalous diffusion processes, such as the observation start time and stepsize when measuring mean square displacements, so as to preclude potentially misleading results and paradoxical interpretations.

Space–time fractional diffusion equations in d-dimensions

We analyze a new space–time fractional diffusion equation encompassing different diffusion processes in d-dimensions. The first-order time derivative is replaced with a time derivative of the Caputo type of arbitrary order β; the spatial-fractional operator of Riesz–Feller or Riesz–Weyl type is replaced with its extension to d-dimensions, defined by means of an extended Fourier transform. The mathematical problem with the spatial-fractional operator proposed here is formulated to tackle anomalous diffusion in heterogeneous media (fractal structures) and incorporating power-law distributions. A formal solution is proposed using the Green function method which, for appropriate initial and boundary conditions, can be expressed in terms of the generalized H-function of Fox—a typical track of anomalous diffusive processes. These mathematical tools provide a new powerful framework to model anomalous diffusion and relaxation problems in heterogeneous media.

Time-averaged mean squared displacement ratio test for Gaussian processes with unknown diffusion coefficient.

The time-averaged mean squared displacement (TAMSD) is one of the most common statistics used for the analysis of anomalous diffusion processes. Anomalous diffusion is manifested by non-linear (mostly power-law) characteristics of the process in contrast to normal diffusion where linear characteristics are expected. One can distinguish between sub- and super-diffusive processes. We consider Gaussian anomalous diffusion models and propose a new approach used for their testing. This approach is based on the TAMSD ratio statistic for different time lags. Similar to the TAMSD, this statistic exhibits a specific behavior in the anomalous diffusion regime. Through its structure, it is independent of the diffusion coefficient, which, in general, does not influence anomalous diffusion behavior. Thus, the TAMSD ratio-based approach does not require preliminary knowledge of the diffusion coefficient's value, in contrast to the TAMSD-approach, where this value is crucial in the testing procedure. Based on the quadratic form representation of the TAMSD ratio, we calculate its main characteristics and propose a step-by-step testing procedure that can be applied for any Gaussian process. For the anomalous diffusion model used here, namely, the fractional Brownian motion, we demonstrate the effectiveness of the proposed methodology. We show that the new approach outperforms the TAMSD-based one, especially for small sample sizes. Finally, the methodology is applied to the real data from the financial market.

Generalized fractional diffusion equation with arbitrary time varying diffusivity

Anomalous diffusion processes in many complex systems are frequently described by the diffusion equation with a time-dependent diffusion coefficient. This paper introduces an exact solution to the broad classes of the fractional diffusion equation with the arbitrary time-dependent diffusion coefficient by using the Laplace-Fourier technique. The Riesz fractional derivative serves to replace the Laplacian operator, while the new regularized Caputo-type fractional derivative is employed instead of the time derivative. We examine our results by introducing and analyzing the most three common cases that represent diffusivity that varying with time. Our calculation shows exact matching with the probability distribution function and mean square displacement illustrated in the literature.

Classification, inference and segmentation of anomalous diffusion with recurrent neural networks

Countless systems in biology, physics, and finance undergo diffusive dynamics. Many of these systems, including biomolecules inside cells, active matter systems and foraging animals, exhibit anomalous dynamics where the growth of the mean squared displacement with time follows a power law with an exponent that deviates from 1. When studying time series recording the evolution of these systems, it is crucial to precisely measure the anomalous exponent and confidently identify the mechanisms responsible for anomalous diffusion. These tasks can be overwhelmingly difficult when only few short trajectories are available, a situation that is common in the study of non-equilibrium and living systems. Here, we present a data-driven method to analyze single anomalous diffusion trajectories employing recurrent neural networks, which we name RANDI. We show that our method can successfully infer the anomalous exponent, identify the type of anomalous diffusion process, and segment the trajectories of systems switching between different behaviors. We benchmark our performance against the state-of-the art techniques for the study of single short trajectories that participated in the Anomalous Diffusion (AnDi) challenge. Our method proved to be the most versatile method, being the only one to consistently rank in the top 3 for all tasks proposed in the AnDi challenge.

Numerical Simulation of an Anomalous Diffusion Process Based on a Scheme of a Higher Order of Accuracy

Numerical Simulation of an Anomalous Diffusion Process Based on a Scheme of a Higher Order of Accuracy

Probability density of fractional Brownian motion and the fractional Langevin equation with absorbing walls

Fractional Brownian motion and the fractional Langevin equation are models of anomalous diffusion processes characterized by long-range power-law correlations in time. We employ large-scale computer simulations to study these models in two geometries, (i) the spreading of particles on a semi-infinite domain with an absorbing wall at one end and (ii) the stationary state on a finite interval with absorbing boundaries at both ends and a source in the center. We demonstrate that the probability density and other properties of the fractional Langevin equation can be mapped onto the corresponding quantities of fractional Brownian motion driven by the same noise if the anomalous diffusion exponent α is replaced by 2 − α. In contrast, the properties of fractional Brownian motion and the fractional Langevin equation with reflecting boundaries were recently shown to differ from each other qualitatively. Specifically, we find that the probability density close to an absorbing wall behaves as P(x) ∼ x κ with the distance x from the wall in the long-time limit. In the case of fractional Brownian motion, κ varies with the anomalous diffusion exponent α as κ = 2/α − 1, as was conjectured previously. We also compare our simulation results to a perturbative analytical approach to fractional Brownian motion.

Numerical study of fast transient non-diffusive heat conduction in a porous medium composed of solid-glass spheres and air using fractional Cattaneo subdiffusion model

This research aims to carry out the fast-transient process of heat flow in a porous medium, proposing the Caputo-type fractional Cattaneo subdiffusion model to study the anomalous diffusion process in the temperature response of the two-phase system. It has been shown that during the short thermal disturbance of porous medium, the solid-fluid interactions in microscale, energy swap through the solid-glass spheres and air, the local non-equilibrium state, and also the fast transient process can be simulated by use of fractional calculus approach. The classical Fourier model fails to interpret such effects, because it is assumed that the thermal wave propagates with infinite speed, and also the thermal interactions between phases occur simultaneously. The energy equation has been developed upon the Caputo fractional derivative, then numerically solved and it has been observed that its results are comparable to the measured data, which reveals the accuracy of the time-fractional model proposed. The numerical results exactly follow the experimental data, which illustrates the applicability of fractional Cattaneo subdiffusion model for examining the transient process in a porous medium at the short temporal scales, and also its capability to interpret the solid-fluid interactions in microscale.

Numerical solution of two-term time-fractional PDE models arising in mathematical physics using local meshless method

Abstract Multi-term time-fractional partial differential equations (PDEs) have become a hot topic in the field of mathematical physics and are used to improve the modeling accuracy in the description of anomalous diffusion processes compared to the single-term PDE results. This research includes the numerical solutions of two-term time-fractional PDE models using an efficient and accurate local meshless method. Due to the advantages of the meshless nature and ease of applicability in higher dimensions, the demand for meshless techniques is increasing. This approach approximates the solution on a uniform or scattered set of nodes, resulting in well-conditioned and sparse coefficient matrices. Numerical tests are performed to demonstrate the efficacy and accuracy of the proposed local meshless technique.

Numerical study of anomalous diffusion of light in semicrystalline polymer structures

From the spread of pollutants in the atmosphere to the transmission of nutrients across cell membranes, anomalous diffusion processes are ubiquitous in natural systems. The ability to understand and control the mechanisms guiding such processes across various scales has important application to research in materials science, finance, medicine, and energetics. Here we present a numerical study of anomalous diffusion of light through a semi-crystalline polymer structure where transport is guided by random disorder and nonlocal interactions. The numerical technique examines diffusion properties in one-dimensional (1D) space via the spectrum of an Anderson-type Hamiltonian with a discrete fractional Laplacian operator (-{\Delta})^s, 0<s<2 and a random distribution of disorder. The results show enhanced transport for s<1 (super-diffusion) and enhanced localization for s>1 (sub-diffusion) for most examined cases. An important finding of the present study is that transport can be enhanced at key spatial scales in the sub-diffusive case, where all states are normally expected to be localized for a (1D) disordered system.

Fractional two energy groups matrix representation for nuclear reactor dynamics with an external source

Nuclear reactor dynamics is the study of the time-dependent behavior of neutron density in an exceeding reactor. Knowledge of however the density in a reactor, which can respond to numerous perturbations, is important to a safe and economical style. Unfortunately, this information is extremely tough to get. To study the behavior of nuclear reactors it is necessary to solve the time-dependent two energy groups neutron diffusion model which is a system of stiff coupled partial differential equations. The two energy groups point kinetics equations with delayed neutrons derived from the neutron diffusion equations in the presence of the time-dependent external neutron source is solved analytically. Since the fractional order differential operator is non-local, then the fractional model of the two energy groups point kinetics equations can be a good representation for the neutron density in the nuclear reactors. In this work, the two energy groups fractional point kinetics equations with multi-group of delayed neutron precursors (2EFPKE) are formulated in the matrix form and two techniques are proposed to solve this system of differential equation analytically, which based on Laplace transformation and eigenvalues and corresponding eigenvectors of the coefficient matrix. The analytical procedure to the proposed fractional model in nuclear reactor dynamics is described and investigated for different types of time-varying reactivity. The performance of developed methods has been tested for both fast and thermal neutron density in the case of the time-dependent external neutron source. In addition, the anomalous diffusion processes, sub-diffusion, and super-diffusion processes are discussed for various fractional order.

Empirical anomaly measure for finite-variance processes

Anomalous diffusion phenomena are observed in many areas of interest. They manifest themselves in deviations from the laws of Brownian motion (BM), e.g. in the non-linear growth (mostly power-law) in time of the ensemble average mean squared displacement (MSD). When we analyze the real-life data in the context of anomalous diffusion, the primary problem is the proper identification of the type of the anomaly. In this paper, we introduce a new statistic, called empirical anomaly measure (EAM), that can be useful for this purpose. This statistic is the sum of the off-diagonal elements of the sample autocovariance matrix for the increments process. On the other hand, it can be represented as the convolution of the empirical autocovariance function with time lags. The idea of the EAM is intuitive. It measures dependence between the ensemble-averaged MSD of a given process from the ensemble-averaged MSD of the classical BM. Thus, it can be used to measure the distance between the anomalous diffusion process and normal diffusion. In this article, we prove the main probabilistic characteristics of the EAM statistic and construct the formal test for the recognition of the anomaly type. The advantage of the EAM is the fact that it can be applied to any data trajectories without the model specification. The only assumption is the stationarity of the increments process. The complementary summary of the paper constitutes of Monte Carlo simulations illustrating the effectiveness of the proposed test and properties of EAM for selected processes.

Log-Normal Superstatistics for Brownian Particles in a Heterogeneous Environment

Superstatistical approaches have played a crucial role in the investigations of mixtures of Gaussian processes. Such approaches look to describe non-Gaussian diffusion emergence in single-particle tracking experiments realized in soft and biological matter. Currently, relevant progress in superstatistics of Gaussian diffusion processes has been investigated by applying χ2-gamma and χ2-gamma inverse superstatistics to systems of particles in a heterogeneous environment whose diffusivities are randomly distributed; such situations imply Brownian yet non-Gaussian diffusion. In this paper, we present how the log-normal superstatistics of diffusivities modify the density distribution function for two types of mixture of Brownian processes. Firstly, we investigate the time evolution of the ensemble of Brownian particles with random diffusivity through the analytical and simulated points of view. Furthermore, we analyzed approximations of the overall probability distribution for log-normal superstatistics of Brownian motion. Secondly, we propose two models for a mixture of scaled Brownian motion and to analyze the log-normal superstatistics associated with them, which admits an anomalous diffusion process. The results found in this work contribute to advances of non-Gaussian diffusion processes and superstatistical theory.

Measurement instrumentation and selected signal processing techniques for anomalous diffusion analysis

Anomalous diffusion phenomena are ubiquitous in many areas, including physics, biology, medicine, climate and also financial markets. The description and analysis of this phenomena are challenging tasks. To fully define the anomalous diffusion and properly understand its nature there is need to combine the skills in advanced sensing techniques, mathematical and statistical knowledge of the corresponding theoretical processes and the experience in physical interpretation of this phenomena. The paper reviews on the measurement instrumentation and signal processing used for detection and description of anomalous diffusion regimes. What is more, estimation of the anomalous diffusion exponent is commented. We present here four methods based on the statistics commonly used in the analyzed problem. Finally the fractional Brownian motion (FBM) is proposed as an analog proper for reliable depiction of properties valid for observed anomalous diffusion phenomena. Simulation studies exhibit the scenario of Hurst exponent estimation in the FBM model when the additive noise of various amplitudes disturbance is added to a simulated trajectory of individual particle. Proposed algorithms have different efficiency depending on the model parameters. This implicates a need to provide more advanced analysis and introduce new methods that could be less sensitive to the measurement noise and/or external forces of noise character. This paper is the starting point for the complete description of the anomalous diffusive processes. It presents the areas for prospective applications of the original sensing techniques but also highlights the needs in a range of statistical and intelligent signal processing suitable for description of the small scale complex systems during the SPT experiment.

Long-time asymptotics of non-degenerate non-linear diffusion equations

We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein’s random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson’s Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation’s solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

A Semi-Analytical Fractal-Fractional Mathematical Model for Multi-Fractured Horizontal Wells in Coalbed Methane Reservoirs

Abstract Coalbed methane (CBM) which is clean energy has received great emphasis recently, and the multi-fracturing technology is widely applied in the exploitation of CBM. Due to the complexity, the randomness, and the anisotropism of the porous medium and the anomalous diffusion process, the fractal theory and fractional calculus are utilized to establish a semi-analytical fractal-fractional mathematical model considering the stress sensitivity of the cleat system for multi-fractured horizontal wells in CBM reservoirs. Through line-sink theory, Pedrosa transformation, perturbation theory, Laplace transformation, element discretization, superposition principle, and Stehfest numerical inversion, the pressure-transient analysis curves are plotted in the double logarithmic coordinates. By comparing with the existing model, the validation of the proposed model is illustrated. Also, nine flowing stages are identified according to different characteristics. Then, sensitivity analysis is conducted and influence laws are summarized. At last, a field application is introduced to furtherly verify the reliability of the proposed model. The relevant results analysis can provide some new significant guidance for interpreting the field data more precisely.

Comb Model with Non-Static Stochastic Resetting and Anomalous Diffusion

Nowadays, the stochastic resetting process is an attractive research topic in stochastic process. At the same time, a series of researches on stochastic diffusion in complex structures introduced ways to understand the anomalous diffusion in complex systems. In this work, we propose a non-static stochastic resetting model in the context of comb structure that consists of a structure formed by backbone in x axis and branches in y axis. Then, we find the exact analytical solutions for marginal distribution concerning x and y axis. Moreover, we show the time evolution behavior to mean square displacements (MSD) in both directions. As a consequence, the model revels that until the system reaches the equilibrium, i.e., constant MSD, there is a Brownian diffusion in y direction, i.e., ⟨ ( Δ y ) 2 ⟩ ∝ t , and a crossover between sub and ballistic diffusion behaviors in x direction, i.e., ⟨ ( Δ x ) 2 ⟩ ∝ t 1 2 and ⟨ ( Δ x ) 2 ⟩ ∝ t 2 respectively. For static stochastic resetting, the ballistic regime vanishes. Also, we consider the idealized model according to the memory kernels to investigate the exponential and tempered power-law memory kernels effects on diffusive behaviors. In this way, we expose a rich class of anomalous diffusion process with crossovers among them. The proposal and the techniques applied in this work are useful to describe random walkers with non-static stochastic resetting on comb structure.

Anomalous diffusion and sorption-desorption process in complex fluid systems

Diffusion processes occurring in a myriad of systems sparkle great interest in understanding their general properties and applications. In this work, we investigate a broad set of diffusive systems that can be governed by a generalized diffusion equation and subjected to a surface that can promote sorption and, consequently, desorption, thus releasing the particles to the bulk. The general bulk equation used here can reproduce different diffusive regimes, among them, those described by the Cattaneo equation or by a fractional, anomalous diffusion. The equation related to the processes on the surface incorporates non-Debye relaxations which can be used to model non-exponential relaxations commonly found in biological or fractal systems. The solutions are obtained by using the Green function approach and show a rich class of behavior that can be related to anomalous diffusion.

Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential.

An important physical property for a stochastic process is how it responds to an external force or spatial confinement. This paper aims to study the relaxation dynamics of a generic process confined in a harmonic potential. We find the dependence of ensemble- and time-averaged mean squared displacements of the confined process on the velocity correlation function C(t,t+τ) of the original process without any external force. Combining two kinds of scaling forms of C(t,t+τ) for small τ and large τ, the stationary value and the relaxation behaviors can be immediately obtained. Our results are valid for a large amount of anomalous diffusion processes, including the ones with single-scaled velocity correlation function (such as fractional Brownian motion and scaled Brownian motion) and the multiscaled ones (like Lévy walk with a broad range of power law exponents of flight time distribution). Note that the latter includes a special case with telegraphic active noise, which could take up athermal energy from the environment.