Continuous Time Random Walk Model Research Articles

An $$\alpha $$-Robust Semidiscrete Finite Element Method for a Fokker–Planck Initial-Boundary Value Problem with Variable-Order Fractional Time Derivative

A time-fractional initial-boundary value problem of Fokker–Planck type is considered on the space-time domain $$\Omega \times [0,T]$$ , where $$\Omega $$ is an open bounded domain in $$\mathbb {R}^d$$ for some $$d\ge 1$$ , and the order $$\alpha (x)$$ of the Riemann-Liouville fractional derivative may vary in space with $$1/2< \alpha (x) < 1$$ for all x. Such problems appear naturally in the formulation of certain continuous-time random walk models. Uniqueness of any solution u of the problem is proved under reasonable hypotheses. A semidiscrete numerical method, using finite elements in space to yield a solution $$u_h(t)$$ , is constructed. Error estimates for $$\Vert (u - u_h)(t)\Vert _{L^2(\Omega )}$$ and $$\int _0^t \left| \partial _t^{1-\alpha } (u-u_h)(s)\right| _1^2 \,ds$$ are proved for each $$t\in [0,T]$$ under the assumptions that the following quantities are finite: $$\Vert u(\cdot , 0)\Vert _{H^2(\Omega )}, |u(\cdot , t)|_{H^1(\Omega )}$$ for each t, and $$\int _0^t [\Vert u(\cdot , t)\Vert _{H^2(\Omega )}^2 + |\partial _t^{1-\alpha }u|_{H^2(\Omega )}^2]$$ , where u(x, t) is the unknown solution. Furthermore, these error estimates are $$\alpha $$ -robust: they do not fail when $$\alpha \rightarrow 1$$ , the classical Fokker–Planck problem. Sharper results are obtained for the special case where the drift term of the problem is not present (which is of interest in certain applications).

SOLUTE TRANSPORT MODELLING IN LOW-PERMEABILITY HOMOGENEOUS AND SATURATED SOIL MEDIA

Fickian and non-Fickian behaviors were often detected for contaminant transport activity owed to the preferential flow and heterogeneity of soil media. Therefore, using diverse methods to measure such composite solute transport in soil media has become an important research topic for solute transport modeling in soil media. In this article, the continuous-time random walk (CTRW) model was applied to illustrate the relative concentration of transport in low-permeability homogeneous and saturated soil media. The solute transport development was also demonstrated with the convection-dispersion equation (CDE) and Two Region Model (TRM) for comparison. CXTFIT 2.1 software was used for CDE and TRM, and CTRW Matlab Toolbox v.3.1 for the CTRW simulation of the breakthrough curve. It was found that higher values of determination coefficient (R2) and lower values of root mean square error (RMSE) concerning the best fits of CDE, TRM, and CTRW. It was found that in the comparison of CDE, TRM, and CTRW, we tend to use CTRW to describe the transport behavior well because there are prevailing Fickian and non-Fickian transport. The CTRW gives better fitting results to the breakthrough curves (BTCs) when β has an increasing pattern towards 2.00. In this study, the variation of parameters in three methods was investigated and results showed that the CTRW modeling approach is more effective to determine non-reactive contaminants concentration in low-permeability soil media at small depths.

Correlated continuous-time random walk in the velocity field: the role of velocity and weak asymptotics.

Within the framework of a space-time correlated continuous-time random walk model, anomalous diffusion of particles moving in the velocity field is studied in this paper. The weak asymptotic form ω(t) ∼ t-(1+α), 1 < α < 2 for large t, is considered to be the waiting time distribution. The analytical results reveal that the diffusion in the velocity field, i.e., the mean squared displacement, can display a multi-fractional form caused by dispersive bias and space-time correlation. The numerical results indicate that the multi-fractional diffusion leads to a crossover phenomenon in-between the process at an intermediate timescale, followed by a steady state which is always determined by the largest diffusion exponent term. In addition, the role of velocity and weak asymptotics is discussed. The extremely small fluid velocity can characterize the diffusion by a diffusion coefficient instead of diffusion exponent, which is distinctly different from the former definition. In particular, for the waiting time displaying a weak asymptotic property, if the anomalous part is suppressed by the normal part, a second crossover phenomenon appears at an intermediate timescale, followed by a steady normal diffusion, which implies that the anomalies underlying the process are smoothed out at large timescales. Moreover, we discuss that the consideration of bias and correlation could help to avoid a possible not readily noticeable mistake in studying the topic concerned in this paper, which may be helpful in the relevant experimental research.

Mean-field tricritical polymers

We provide an introductory account of a tricritical phase diagram, in the setting of a mean-field random walk model of a polymer density transition, and clarify the nature of the density transition in this context. We consider a continuous-time random walk model on the complete graph, in the limit as the number of vertices $N$ in the graph grows to infinity. The walk has a repulsive self-interaction, as well as a competing attractive self-interaction whose strength is controlled by a parameter $g$. A chemical potential $\nu$ controls the walk length. We determine the phase diagram in the $(g,\nu)$ plane, as a model of a density transition for a single linear polymer chain. A dilute phase (walk of bounded length) is separated from a dense phase (walk of length of order $N$) by a phase boundary curve. The phase boundary is divided into two parts, corresponding to first-order and second-order phase transitions, with the division occurring at a tricritical point. The proof uses a supersymmetric representation for the random walk model, followed by a single block-spin renormalisation group step to reduce the problem to a 1-dimensional integral, followed by application of the Laplace method for an integral with a large parameter.

Large Deviations for Continuous Time Random Walks.

Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

Fractional-order heat conduction models from generalized Boltzmann transport equation.

The relationship between fractional-order heat conduction models and Boltzmann transport equations (BTEs) lacks a detailed investigation. In this paper, the continuity, constitutive and governing equations of heat conduction are derived based on fractional-order phonon BTEs. The underlying microscopic regimes of the generalized Cattaneo equation are thereafter presented. The effective thermal conductivity κeff converges in the subdiffusive regime and diverges in the superdiffusive regime. A connection between the divergence and mean-square displacement 〈|Δx|2〉 ∼ tγ is established, namely, κeff ∼ tγ-1, which coincides with the linear response theory. Entropic concepts, including the entropy density, entropy flux and entropy production rate, are studied likewise. Two non-trivial behaviours are observed, including the fractional-order expression of entropy flux and initial effects on the entropy production rate. In contrast with the continuous time random walk model, the results involve the non-classical continuity equations and entropic concepts. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.

Experimental and modeling evidence of kilometer-scale anomalous tracer transport in an alpine karst aquifer

Karst aquifers are important drinking water resources, but highly vulnerable to contamination. Contaminants can be transported rapidly through a network of fractures and conduits, with only limited sorption or degradation, which usually leads to a fast and strong response at karst springs. During migration, contaminants can also enter less mobile zones, such as pools or water in intra-karstic sediments, or advance from conduits into the adjacent fractured rock matrix. As contaminant concentrations in the main flow path(s) decrease, contaminants may migrate back into the main flow path and reach the karst springs at low (but significant) concentrations over a long time span. This is the conventional interpretation for the oft-observed steep rising limb and the long-tailed falling limb of tracer breakthrough curves in karst systems. Here, field measurements are examined from an alpine karst system in Austria where a series of distinctive, long-tailed breakthrough curves (BTCs) of conservative tracers were observed over distances up to 7400 m. Recognizing that the conventional advection-dispersion equation (ADE) cannot usually quantify such behavior, two other modeling approaches are considered, namely the two-region non-equilibrium (2RNE) model, which explicitly includes mobile and immobile zones, and a continuous time random walk (CTRW) model, which is based on a physically-based, probabilistic approach that describes anomalous (or non-Fickian) transport behavior characteristic of heterogeneous systems such as karst. In most cases, the ADE and 2RNE models do not quantify the low concentrations at longer travel times. The CTRW, in contrast, accounts for the long-tailed breakthrough behavior found in this karst system.

Observation time dependent mean first passage time of diffusion and subdiffusion processes

The mean first passage time, one of the important characteristics for a stochastic process, is often calculated assuming the observation time is infinite. However, in practice, the observation time, T, is always finite and the mean first passage time (MFPT) is dependent on the length of the observation time. In this work, we investigate the observation time dependence of the MFPT of a particle freely moving in the interval [−L,L] for a simple diffusion model and five different models of subdiffusion, the fractional diffusion equation (FDE), scaled Brown motion (SBM), fractional Brownian motion (FBM), stationary Markovian approximation of SBM, and the aging continuous-time random walk model. We find that the MFPT is linearly dependent on T in the small T limit for all the models investigated, while the large-T behavior of the MFPT is sensitive to stochastic properties of the transport model in question. We also discuss the relationship between the observation time, T, dependence and the travel length, L, dependence of the MFPT. Our results suggest the observation time dependency of the MFPT can serve as an experimental measure that is far more sensitive to stochastic properties of transport processes than the mean square displacement.

Quasi-diffusion magnetic resonance imaging (QDI): A fast, high b-value diffusion imaging technique

To enable application of non-Gaussian diffusion magnetic resonance imaging (dMRI) techniques in large-scale clinical trials and facilitate translation to clinical practice there is a requirement for fast, high contrast, techniques that are sensitive to changes in tissue structure which provide diagnostic signatures at the early stages of disease. Here we describe a new way to compress the acquisition of multi-shell b-value diffusion data, Quasi-Diffusion MRI (QDI), which provides a probe of subvoxel tissue complexity using short acquisition times (1–4 ​min). We also describe a coherent framework for multi-directional diffusion gradient acquisition and data processing that allows computation of rotationally invariant quasi-diffusion tensor imaging (QDTI) maps.QDI is a quantitative technique that is based on a special case of the Continuous Time Random Walk model of diffusion dynamics and assumes the presence of non-Gaussian diffusion properties within tissue microstructure. QDI parameterises the diffusion signal attenuation according to the rate of decay (i.e. diffusion coefficient, D in mm2 s−1) and the shape of the power law tail (i.e. the fractional exponent, α). QDI provides analogous tissue contrast to Diffusional Kurtosis Imaging (DKI) by calculation of normalised entropy of the parameterised diffusion signal decay curve, Hn, but does so without the limitations of a maximum b-value.We show that QDI generates images with superior tissue contrast to conventional diffusion imaging within clinically acceptable acquisition times of between 84 and 228 ​s. We show that QDI provides clinically meaningful images in cerebral small vessel disease and brain tumour case studies. Our initial findings suggest that QDI may be added to routine conventional dMRI acquisitions allowing simple application in clinical trials and translation to the clinical arena.

Simplified calibration of continuous-time random walk solute transport models

Continuous-time random walk (CTRW) models of non-Fickian solute transport are defined by a temporal pdf and three parameters: a velocity-like constant, a dispersion-like constant and a “time constant”. Presently, to identify a model, all are jointly calibrated against solute breakthrough data. We show that without loss of generality the time constant can be set to unity and velocity-like and dispersion-like CTRW parameters can be set equal to well-defined classical counterparts, and thus physically pre-constrained. Thus only one entity, the pdf, requires empirical fitting during model calibration, rather than four. Our assertions are backed by numerical analysis.

A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative

We provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The equation is derived by continuous time random walk model, being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with L2-norm and mathematical induction with l∞ norm). Moreover, the optimal estimates are obtained.

Comparison of negative skewed space fractional models with time nonlocal approaches for stream solute transport modeling

Continuous time random walks (CTRW), multi-rate mass transfer (MRMT), and fractional advection-dispersion equations (FADEs) are three promising models of anomalous transport as commonly found in natural streams. Although these paradigms are mathematically related, understanding their advantages and limitations poses a challenge for model selection. In this paper, we quantitatively evaluate the advection-dispersion equation (ADE), fractional-mobile-immobile (FMIM), fractional-in-space ADE (sFADE), fractional in space transient storage (FSTS), truncated time-fractional model (TTFM), and CTRW models with truncated power-law waiting time distribution (CTRW-TPL) by fitting them first to synthetic data. We then applied these models to observations from tracer experiments conducted in several rivers. Based on the extensive analysis, we conclude that the performance of the FSTS model (in particular, the model with negative skewness β=-1) is comparable or superior to the other nonlocal models evaluated in the paper; therefore, the model represents an alternative to existing models for simulating stream solute transport for spatially-homogeneous flows.

An investigation on continuous time random walk model for bedload transport

Abstract Bedload particles in the armoring layer may experience a multi-scale effect and multiple mass transfer rates between mobile and immobile domains. Anomalous transport behaviors and retarded space evolution plume cannot be described by the normal diffusion equation. In this paper, we apply the continuous time random walk model with different distributions of waiting times to capture bedload transport behavior under different conditions. Experimental data indicate that fluctuations of diffusive rates for bedload transport can be captured by the truncated power law (TPL). The retarded plume evolution can be well characterized by an exponential distribution of waiting times and advection-diffusion equation with a retarded kernel. The heavy-tailed snapshots of bedload transport are interpreted in terms of mobile and immobile states.

Impact of bacteria motility in the encounter rates with bacteriophage in mucus

Bacteriophages—or phages—are viruses that infect bacteria and are present in large concentrations in the mucosa that cover the internal organs of animals. Immunoglobulin (Ig) domains on the phage surface interact with mucin molecules, and this has been attributed to an increase in the encounter rates of phage with bacteria in mucus. However, the physical mechanism behind this phenomenon remains unclear. A continuous time random walk (CTRW) model simulating the diffusion due to mucin-T4 phage interactions was developed and calibrated to empirical data. A Langevin stochastic method for Escherichia coli (E. coli) run-and-tumble motility was combined with the phage CTRW model to describe phage-bacteria encounter rates in mucus for different mucus concentrations. Contrary to previous theoretical analyses, the emergent subdiffusion of T4 in mucus did not enhance the encounter rate of T4 against bacteria. Instead, for static E. coli, the diffusive T4 mutant lacking Ig domains outperformed the subdiffusive T4 wild type. E. coli’s motility dominated the encounter rates with both phage types in mucus. It is proposed, that the local fluid-flow generated by E. coli’s motility combined with T4 interacting with mucins may be the mechanism for increasing the encounter rates between the T4 phage and E. coli bacteria.

Wind turbine load dynamics in the context of turbulence intermittency

Abstract. The importance of a high-order statistical feature of wind, which is neglected in common wind models, is investigated: non-Gaussian distributed wind velocity increments related to the intermittency of turbulence and their impact on wind turbine dynamics and fatigue loads are the focus. Gaussian and non-Gaussian synthetic wind fields obtained from a continuous-time random walk model are compared and fed to a common aero-servo-elastic model of a wind turbine employing blade element momentum (BEM) aerodynamics. It is discussed why and how the effect of the non-Gaussian increment statistics has to be isolated. This is achieved by assuring that both types feature equivalent probability density functions, spectral properties and coherence, which makes them indistinguishable based on wind characterizations of common design guidelines. Due to limitations in the wind field genesis, idealized spatial correlations are considered. Three examples with idealized; differently sized wind structures are presented. A comparison between the resulting wind turbine loads is made. For the largest wind structure sizes, differences in the fatigue loads between intermittent and Gaussian are observed. These are potentially relevant in a wind turbine certification context. Subsequently, the dependency of this intermittency effect on the field's spatial variation is discussed. Towards very small structured fields, the effect vanishes.

Spontaneous Magnetic Superdomain Wall Fluctuations in an Artificial Antiferromagnet.

Collective dynamics often play an important role in determining the stability of ground states for both naturally occurring materials and metamaterials. We studied the temperature dependent dynamics of antiferromagnetically ordered superdomains in a square artificial spin lattice using soft x-ray photon correlation spectroscopy. We observed an exponential slowing down of superdomain wall motion below the antiferromagnetic onset temperature, similar to the behavior of typical bulk antiferromagnets. Using a continuous time random walk model we show that these superdomain walls undergo low-temperature ballistic and high-temperature diffusive motions.

On Modeling Ensemble Transport of Metal Reducing Motile Bacteria

Many metal reducing bacteria are motile with their run-and-tumble behavior exhibiting series of flights and waiting-time spanning multiple orders of magnitude. While several models of bacterial processes do not consider their ensemble motion, some models treat motility using an advection diffusion equation (ADE). In this study, Geobacter and Pelosinus, two metal reducing species, are used in micromodel experiments for study of their motility characteristics. Trajectories of individual cells on the order of several seconds to few minutes in duration are analyzed to provide information on (1) the length of runs, and (2) time needed to complete a run (waiting or residence time). A Continuous Time Random Walk (CTRW) model to predict ensemble breakthrough plots is developed based on the motility statistics. The results of the CTRW model and an ADE model are compared with the real breakthrough plots obtained directly from the trajectories. The ADE model is shown to be insufficient, whereas a coupled CTRW model is found to be good at predicting breakthroughs at short distances and at early times, but not at late time and long distances. The inadequacies of the simple CTRW model can possibly be improved by accounting for correlation in run length and waiting time.

Continuous time random walk model for non-uniform bed-load transport with heavy-tailed hop distances and waiting times

Bed-load transport along widely graded river-beds typically exhibits anomalous dynamics, whose efficient characterization may require parsimonious stochastic models with pre-defined statistics involving the waiting time and hop distance distributions for sediment particles. This study employs a continuous time random walk (CTRW) model to characterize bed-load particle motions on a widely graded gravel-bed with cluster microforms built in our lab. Flume experiments guide the selection of the Mittag-Leffler (M-L) function as the waiting time distribution function, and the Lévy α-stable density for the hop distance distribution function in the CTRW model. Monte Carlo simulations show that the resulting CTRW model can well capture the observed flume experimental data (with either a continuous or an instantaneous source) with coexisting super- and sub-dispersion behaviors in the bed-load transport process. Analyses further discover the dual impact of clusters on the dynamics of fine sediment particles. Some particles are more likely to be blocked or trapped by clusters, while others have a high probability to be accelerated by the flow accelerating belt between the clusters. Therefore, with proper statistical distributions and relevant parameters for sediment waiting times and hop distances, the CTRW model may efficiently capture the complex dynamics in sediment transport.

Continuous-Time Random Walk for a Particle in a Periodic Potential

Continuous-time random walks offer powerful coarse-grained descriptions of transport processes. We here microscopically derive such a model for a Brownian particle diffusing in a deep periodic potential. We determine both the waiting-time and the jump-length distributions in terms of the parameters of the system, from which we analytically deduce the non-Gaussian characteristic function. We apply this continuous-time random walk model to characterize the underdamped diffusion of single cesium atoms in a one-dimensional optical lattice. We observe excellent agreement between experimental and theoretical characteristic functions, without any free parameter.

Effect of colloids on non-Fickian transport of strontium in sediments elucidated by continuous-time random walk analysis

Understanding the influence of colloids on radionuclide migration is of significance to evaluate environmental risks for radioactive waste disposals. In order to formulate an appropriate modelling framework that can quantify and interpret the anomalous transport of Strontium (Sr) in the absence and presence of colloids, the continuous time random walk (CTRW) approach is implemented in this work using available experimental information. The results show that the transport of Sr and its recovery are enhanced in the presence of colloids. The causes can be largely attributed to the trap-release processes, e.g. electrostatic interactions of Sr, colloids and natural sediments, and differences in pore structures, which gave rise to the varying interstitial velocities of dissolved and, if any, colloid-associated Sr. Good agreement between the CTRW simulations and the column-scale observations is demonstrated. Regardless of the presence of colloids, the CTRW modelling captures the characteristics of non-Fickian anomalous transport (0 < β < 2) of Sr. In particular, a range of 0 < β < 1, corresponding to the cases with greater recoveries, reveal strongly non-Fickian transport with distinctive earlier arrivals and tailing effects, likely due to the physicochemical heterogeneities, i.e. the repulsive interactions and/or the macro-pores originating from local heterogeneities. The results imply that colloids can increase the Sr transport as a barrier of Sr sorption onto sediments herein, apart from often being carriers of sored radionuclides in aqueous phase. From a modelling perspective, the findings show that the established CTRW model is valid for quantifying the non-Fickian and promoted transport of Sr with colloids.