Continuous Time Random Walk Theory Research Articles

Stochastic modeling of injection induced seismicity based on the continuous time random walk model

The spatiotemporal evolution of earthquakes induced by fluid injections into the subsurface can be erratic owing to the complexity of the physical process. To effectively mitigate the associated hazard and to draft appropriate regulatory strategies, a detailed understanding of how induced seismicity may evolve is needed. In this work, we build on the well-established continuous-time random walk (CTRW) theory to develop a purely stochastic framework that can delineate the essential characteristics of this process. We use data from the 2003 and 2012 hydraulic stimulations in the Cooper Basin geothermal field that induced thousands of microearthquakes to test and demonstrate the applicability of the model. Induced seismicity in the Cooper Basin shows all the characteristics of subdiffusion, as indicated by the fractional order power-law growth of the mean square displacement with time and broad waiting-time distributions with algebraic tails. We further use an appropriate master equation and the time-fractional diffusion equation to map the spatiotemporal evolution of seismicity. The results show good agreement between the model and the data regarding the peak earthquake concentration close to the two injection wells and the stretched exponential relaxation of seismicity with distance, suggesting that the CTRW model can be efficiently incorporated into induced seismicity forecasting.

Sharp Transition to Strongly Anomalous Transport in Unsaturated Porous Media

AbstractThe simultaneous presence of liquid and gas in porous media increases flow heterogeneity compared to saturated flows. However, the impact of saturation on flow and transport has so far remained unclear. The presence of gas in the pore space leads to flow reorganization. We develop a theoretical framework that captures the impact of that reorganization on pore‐scale fluid velocities. Preferential flow is distributed spatially through a backbone, and flow recirculation occurs in flow dead‐ends. We observe, and predict theoretically, that this previously identified flow structure induces a marked change in the scaling of the velocity probability density function compared to the saturated configuration and a sharp transition to strongly anomalous transport. We develop a transport model using the continuous time random walk theory that predicts advective transport dynamics for all saturation degrees. Our results provide a new modeling framework linking phase heterogeneity to flow heterogeneity in unsaturated porous media.

Active interaction switching controls the dynamic heterogeneity of soft colloidal dispersions.

We employ Reactive Dynamical Density Functional Theory (R-DDFT) and Reactive Brownian Dynamics (R-BD) simulations to investigate the dynamics of a suspension of active soft Gaussian colloids with binary interaction switching, i.e., a one-component colloidal system in which every particle stochastically switches at predefined rates between two interaction states with different mobility. Using R-DDFT we extend a theory previously developed to access the dynamics of inhomogeneous liquids [Archer et al., Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75, 040501] to study the influence of the switching activity on the self and distinct part of the Van Hove function in bulk solution, and determine the corresponding mean squared displacement of the switching particles. Our results demonstrate that, even though the average diffusion coefficient is not affected by the switching activity, it significantly modifies the non-equilibrium dynamics and diffusion coefficients of the individual particles, leading to a crossover from short to long times, with a regime for intermediate times showing anomalous diffusion. In addition, the self-part of the van Hove function has a Gaussian form at short and long times, but becomes non-Gaussian at intermediates ones, having a crossover between short and large displacements. The corresponding self-intermediate scattering function shows the two-step relaxation patters typically observed in soft materials with heterogeneous dynamics such as glasses and gels. We also introduce a phenomenological Continuous Time Random Walk (CTRW) theory to understand the heterogeneous diffusion of this system. R-DDFT results are in excellent agreement with R-BD simulations and the analytical predictions of CTRW theory, thus confirming that R-DDFT constitutes a powerful method to investigate not only the structure and phase behavior, but also the dynamical properties of non-equilibrium active switching colloidal suspensions.

Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction

Blood microcirculation supplies neurons with oxygen and nutrients, and contributes to clearing their neurotoxic waste, through a dense capillary network connected to larger tree-like vessels. This complex microvascular architecture results in highly heterogeneous blood flow and travel time distributions, whose origin and consequences on brain pathophysiology are poorly understood. Here, we analyze highly-resolved intracortical blood flow and transport simulations to establish the physical laws governing the macroscopic transport properties in the brain micro-circulation. We show that network-driven anomalous transport leads to the emergence of critical regions, whether hypoxic or with high concentrations of amyloid-β, a waste product centrally involved in Alzheimer’s Disease. We develop a Continuous-Time Random Walk theory capturing these dynamics and predicting that such critical regions appear much earlier than anticipated by current empirical models under mild hypoperfusion. These findings provide a framework for understanding and modelling the impact of microvascular dysfunction in brain diseases, including Alzheimer’s Disease.

Stochastic model of oscillatory diffusion for colloidal particles in a fixed porous bed

Colloidal particles in the pore spaces within a packed bed move back and forth in response to an imposed oscillatory flow, such as that associated with an acoustic field. The diffusive motion of the particles results from intermittent hindering of the oscillatory motion, which we assume to result from particle filtration by the packed bed pore spaces. We consider an experimentally-validated stochastic model that entails a series of transitions between an oscillatory state, where the particle oscillates with the fluid flow, and a captured state, where the particle is held fixed in the bed. The paper examines sensitivity of this stochastic model to different parameters and compares the stochastic model predictions for diffusion coefficient with predictions of an analytical solution based on continuous-time random walk (CTRW) theory. The results are relevant to applications such as nanoparticle penetration into biofilms, drug capsule penetration into human tissue, and microplastic transport within saturated soil.

Analyzing and modeling sub-diffusive transport of bedload along a heterogeneous gravel bed using stochastic and statistical methods

Bedload transport is one of the main mechanisms for sediment transport in rivers. Bedload transport may exhibit anomalous dispersion behavior during the formation of clusters on the surface of a heterogeneous river bed, which cannot be quantified by the classical Fick’s law of diffusion. This study simplifies the complex bedload transport process to a “mass-spring-damper” system, which can describe the cluster formation process based on the observed cluster geometry. The simulation results are consistent with the morphology of clusters extracted from the experimental gravel bed in our flume. The level of spatial heterogeneity of individual clusters is characterized to distinguish the planform morphology of each cluster, based on estimation of the Hausdorff-Besicovitch dimension of the cluster area. Trapping of sediment particles during bedload transport on a heterogeneous gravel bed is quantified using a random walk approach and the birth-death emigration-immigration Markov process. Sub-diffusive dynamics of bedload transport is also modeled simultaneously using the Langevin equation that defines statistical properties of bedload sediments. Further analysis indicates that the microscopic stochastic Langevin equation relates to a macroscopic deterministic equation according to the continuous time random walk (CTRW) theory.

Anomalous losses of energetic particles in the presence of an oscillating radial electric field in fusion plasmas

The confinement of energetic particles in nuclear fusion devices is studied in the presence of an oscillating radial electric field and an axisymmetric magnetic equilibrium. It is shown that, despite the poloidal and toroidal symmetries, initially integrable orbits turn into chaotic regions that can potentially intercept the wall of the tokamak, leading to particle losses. It is observed that the losses exhibit algebraic time decay different from the expected exponential decay characteristic of radial diffusive transport. A dynamical explanation of this behaviour is presented, within the continuous time random walk theory. The central point of the analysis is based on the fact that, contrary to the radial displacement, the poloidal angle is not bounded and a proper statistical analysis can therefore be made, showing for the first time that energetic particle transport can be super-diffusive in the poloidal direction and characterised by asymmetric poloidal displacement. The connection between poloidal and radial positions ensured by the conservation of the toroidal canonical momentum, implies that energetic particles spend statistically more time in the inner region of the tokamak than in the outer one, which explains the observed algebraic decay. This indicates that energetic particles might be efficiently slowed down by the thermal population before leaving the system. Also, the asymmetric transport reveals a new possible mechanism of self-generation of momentum.

A Survey of Models of Ultraslow Diffusion in Heterogeneous Materials

Ultraslow diffusion is characterized by a logarithmic growth of the mean squared displacement (MSD) as a function of time. It occurs in complex arrangements of molecules, microbes, and many-body systems. This paper reviews mechanical models for ultraslow diffusion in heterogeneous media from both macroscopic and microscopic perspectives. Macroscopic models are typically formulated in terms of a diffusion equation that employs noninteger order derivatives (distributed order, structural, and comb models (CM)) or employs a diffusion coefficient that is a function of space or time. Microscopic models are usually based on the continuous time random walk (CTRW) theory, but use a weighted logarithmic function as the limiting formula of the waiting time density. The similarities and differences between these models are analyzed and compared with each other. The corresponding MSD in each case is tabulated and discussed from the perspectives of the underlying assumptions and of real-world applications in heterogeneous materials. It is noted that the CMs can be considered as a type of two-dimensional distributed order fractional derivative model (DFDM), and that the structural derivative models (SDMs) generalize the DFDMs. The heterogeneous diffusion process model (HDPM) with time-dependent diffusivity can be rewritten to a local structural derivative diffusion model mathematically. The ergodic properties, aging effect, and velocity autocorrelation for the ultraslow diffusion models are also briefly discussed.

Diffusion equations with general nonlocal time and space derivatives

In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book (Bisci et al., 2016).

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

From Power Laws to Fractional Diffusion Processes with and Without External Forces, the Non Direct Way

Abstract In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.

On the time-fractional Cattaneo equation of distributed order

The present work revisits the time-fractional Cattaneo equation (TFCE) by considering a generic class of time-fractional derivatives of distributed order, α∈(0,1]. At first, we shed light on two (main) forms of TFCE; one (natural generalization) uses the Caputo fractional derivative and the other (Compte–Metzler generalization) is defined in the Riemann–Liouville sense, and the equivalence condition between their solutions is provided. Some special cases of these two forms are addressed. Hence, different versions of distributed-order TFCE are proposed. The fundamental solution of the two general forms of distributed-order TFCE are derived using integral transform technique. The non-negativity of solutions of TFCE and its distributed-order versions is examined using the “special-type functions” technique; Gorenflo, Luchko and Stojanović (2013). Two classes of order distribution are considered: discrete-order and continuous-order distributions, and some random choices of non-negative order densities affirm graphically the non-negativity of the general forms of distributed-order TFCE. The double-order (as an example on the discrete-order), and exponentially exp(λα) and sinusoidally sin(πμα) distributed-order (as an example on the continuous-order) are chosen. Finally, connections between some models of distributed-order TFCE and continuous time random walk theory are established, and the corresponding mean-squared displacements are discussed.

A Practical Modeling Framework for Non-Fickian Transport and Multi-Species Sequential First-Order Reaction.

Many studies indicate that small-scale heterogeneity and/or mobile-immobile mass exchange produce transient non-Fickian plume behavior that is not well captured by the use of the standard, deterministic advection-dispersion equation (ADE). An extended ADE modeling framework is presented here that is based on continuous time random walk theory. It can be used to characterize non-Fickian transport coupled with simultaneous sequential first-order reactions (e.g., biodegradation or radioactive decay) for multiple degrading contaminants such as chlorinated solvents, royal demolition explosive, pesticides, and radionuclides. To demonstrate this modeling framework, new transient analytical solutions are derived and are inverted in Laplace space. Closed-form, steady-state, multi-species analytical solutions are also derived for non-Fickian transport in highly heterogeneous aquifers with linear sorption-desorption and matrix diffusion for use in spreadsheets. The solutions are general enough to allow different degradation rates for the mobile and immobile zones. The transient solutions for multi-species transport are applied to examine the effects of source remediation on the natural attenuation of downgradient plumes of both parent and degradation products in highly heterogeneous aquifers. Results for representative settings show that the use of the standard, deterministic ADE can over-estimate cleanup rates and under-predict the cleanup timeframe in comparison to the extended ADE analytical model. The modeling framework and calculations introduced here are also applied for a 30 year groundwater cleanup program at a site in Palm Bay, Florida. The simulated plume concentrations using the extended ADE exhibited agreement with observed long concentration tails of trichloroethene, cis 1,2 DCE, and VC that remained above cleanup goals.

From continuous time random walks to the generalized diffusion equation

Abstract We obtain a generalized diffusion equation in modified or Riemann-Liouville form from continuous time random walk theory. The waiting time probability density function and mean squared displacement for different forms of the equation are explicitly calculated. We show examples of generalized diffusion equations in normal or Caputo form that encode the same probability distribution functions as those obtained from the generalized diffusion equation in modified form. The obtained equations are general and many known fractional diffusion equations are included as special cases.

The fractional Boussinesq equation of groundwater flow and its applications

This paper presents a set of fractional Boussinesq equations (fBEs) for groundwater flow in confined and unconfined aquifers and demonstrates the application of one of the fBEs for groundwater discharges known as recession curves. The fBEs are formulated with two-term distributed fractional orders in time and symmetrical fractional derivatives (SFD) in space applicable to both confined and unconfined aquifers. The SFD in theory consists of the forward fractional derivative (FFD) and the backward fractional derivative (BFD). The FFD represents the forward movement of water along the direction of mainstream flow while the BFD accounts for the backward motion of water in the direction opposite to the mainstream flow. The backward flow at the pore level can be referred to as the micro-scale backwater effect. The analogue of the backwater effect on a micro-scale using the BFD coincides with the wandering processes based on the continuous-time random walk (CTRW) theory which results in the fractional governing equation. With the analytical solutions of the fBE for given initial and boundary conditions of the first type for a finite depth, a set of formulae for groundwater recession has been derived using approximate solutions of the fBE. The examples of the applications of the recession curves are graphically illustrated and the effects of the orders of fractional derivatives on the geometry of the flow curves examined.

Modeling Bromide Transport in Undisturbed Soil Columns with the Continuous Time Random Walk

Nowadays, mathematical models are commonly used as efficient tools in solute transport studies and management in porous media. In this paper, we evaluated the continuous time random walk (CTRW) theory for its capability to characterize bromide transport in one-dimensional saturated undisturbed clay loam and sandy loam soil columns (10 cm in diameter and 40 cm long). The transport process was also simulated by using the advection–dispersion equation (ADE) for comparison. The values of CTRW parameter, β, for clay loam and sandy loam soil columns were derived at 1.71 and 1.73, respectively. This indicated that the bromide transport behavior within the clay loam and sandy loam soil columns was anomalous transport or non-Fickian transport and CTRW model was more suitable for simulation of bromide transport through undisturbed clay loam and sandy soil columns compared to ADE model with Fickian diffusion law basis. The graphical and statistical analysis confirmed the efficiency of CTRW model for simulation of bromide transport.

Parsimonious continuous time random walk models and kurtosis for diffusion in magnetic resonance of biological tissue.

In this paper, we provide a context for the modeling approaches that have been developed to describe non-Gaussian diffusion behavior, which is ubiquitous in diffusion weighted magnetic resonance imaging of water in biological tissue. Subsequently, we focus on the formalism of the continuous time random walk theory to extract properties of subdiffusion and superdiffusion through novel simplifications of the Mittag-Leffler function. For the case of time-fractional subdiffusion, we compute the kurtosis for the Mittag-Leffler function, which provides both a connection and physical context to the much-used approach of diffusional kurtosis imaging. We provide Monte Carlo simulations to illustrate the concepts of anomalous diffusion as stochastic processes of the random walk. Finally, we demonstrate the clinical utility of the Mittag-Leffler function as a model to describe tissue microstructure through estimations of subdiffusion and kurtosis with diffusion MRI measurements in the brain of a chronic ischemic stroke patient.

Effects of benthic and hyporheic reactive transport on breakthrough curves

AbstractIn streams and rivers, the benthic and hyporheic regions harbor the microbes that process many stream-borne constituents, including O2, nutrients, C, and contaminants. The full distribution of transport time scales in these highly reactive regions must be understood because solute delivery and extended storage in these metabolically active zones control the opportunity for biogeochemical processing. The most commonly used transport models cannot capture these effects. We present a stochastic model for conservative and reactive solute transport in rivers based on continuous-time random-walk theory, which is capable of distinguishing and capturing processes not described by classical approaches. The model includes surface and subsurface storage zones with arbitrary residence-time distributions. We used this model to evaluate the effects of sorption and biological uptake on downstream solute transport. Linear or mildly nonlinear sorption in storage delays downstream transport without changing the fun...

Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single particle tracking to demonstrate that the motion of DC-SIGN, a receptor with unique pathogen recognition capabilities, reveals nonergodic subdiffusion on living cell membranes. In contrast to previous studies, this behavior is incompatible with transient immobilization and therefore it can not be interpreted according to continuous time random walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Due to its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology and ecology.

Diffusion in randomly perturbed dissipative dynamics

Dynamical systems having many coexisting attractors present interesting properties from both fundamental theoretical and modelling points of view. When such dynamics is under bounded random perturbations, the basins of attraction are no longer invariant and there is the possibility of transport among them. Here we introduce a basic theoretical setting which enables us to study this hopping process from the perspective of anomalous transport using the concept of a random dynamical system with holes. We apply it to a simple model by investigating the role of hyperbolicity for the transport among basins. We show numerically that our system exhibits non-Gaussian position distributions, power-law escape times, and subdiffusion. Our simulation results are reproduced consistently from stochastic continuous time random walk theory.