Continuous Time Random Walk Research Articles

Solute Trapping and the Mechanisms of Non‐Fickian Transport in Partially Saturated Porous Media

AbstractWe study the upscaling of pore‐scale solute transport in partially saturated porous media at different saturation degrees. The interaction between structural heterogeneity, phases distribution and small‐scale flow dynamics induces complex flow patterns and broad probability distributions of flow, which control key aspects of transport, such as residence and arrival times, dispersion, and spatial solute distributions, as well as chemical reactions. A continuous‐time random walk (CTRW) framework that integrates the processes of advection, diffusion, and trapping in immobile zones is used to upscale and evaluate the transport of diluted solutes. Results of this model were compared to direct numerical simulations solving the advection‐diffusion equation in experimental saturation patterns. The comparison between simulations results, with different Péclet numbers (Pe), and the physics‐based upscaled CTRW approach allows for a quantitative analysis of the governing factors of transport in partially saturated porous media. This analysis shows that the fluid phase saturation decreases the advective tortuosity, the media's characteristic length, the fraction of the immobile region, and the mean trapping time. At the same time, for a given saturation degree, the normalized mean trapping time is proportional to the Pe. This suggests that the characteristic trapping length is proportional to the media's characteristic (correlation) length. Moreover, the trapping frequency decreases with increasing Pe.

Anomalous dielectric relaxation with multispecies linear reaction dynamics

Anomalous dielectric relaxation in complex reactions is difficult to specify, since there is no classic model for the reacting species with anomalous subdiffusion in the presence of an external electric field. In this paper, the anomalous dielectric relaxation of a multispecies system with linear reaction dynamics is studied based on the biased continuous time random walks. Different reaction–subdiffusion equations are derived via the mathematical random subordination time and the deterministic physical time. The mathematical random subordination time is based on the assumption that the jumping probabilities are determined at the end of the waiting times, while the deterministic physical time is based on the assumption that the jumping probabilities are independent of time and evaluated at the start of the waiting times. The force fields are operated by the Riemann–Liouville fractional derivative in the reaction–subdiffusion equations with the deterministic physical time, while the opposite results are found in the reaction–subdiffusion equations with the mathematical random subordination time. The explicit expressions for the anomalous dielectric relaxation of a reversible reaction are further derived according to the two reaction–subdiffusion equations in the presence of space- and time-dependent electric fields. The numerical calculations are carried to validate the reliability of the different expressions for the anomalous dielectric relaxation, and the reaction–subdiffusion equation with the deterministic physical time is proved to be suitable for describing the anomalous dielectric relaxation.

Entanglement transfer via chiral and continuous-time quantum walks on a triangular chain

We investigate the chiral quantum walk (CQW) as a mechanism for an entanglement transfer on a triangular chain structure. We specifically consider two-site spatially entangled cases in short-time and long-time regimes. Using the concurrence as an entanglement measure; fidelity, and the Bures distance as the measure of the quality of the state transfer, we evaluate the success of the entanglement transfer. We compare the entangled state transfer time and quality in CQW against a continuous-time quantum random walk. We also observe the effect of mixed states on the entanglement transfer quality.

Global stability of a fractional order SIS epidemic model

In this paper, we establish a new fractional order SIS (frSIS) model by way of continuous time random walk. The value of this study lies in two aspects. Mathematically, we provide a framework for the global stability of the frSIS model, and prove that the basic reproduction number R0 can be used to govern the dynamics of the frSIS model. If R0<1, the disease-free equilibrium of the model is globally asymptotically stable; if R0>1, the endemic equilibrium of the model is globally asymptotically stable. And epidemiologically, we find that, in order to control the spread of the disease, we must decrease the death rate and the average infectious period to make the disease go to extinction, which can provide us with some useful control strategies to regulate disease dynamics.

A generalised diffusion equation corresponding to continuous time random walks with coupling between the waiting time and jump length distributions

This work outlines a transiently coupled continuous time random walk framework. The coupling is between the displacement probability density function (PDF) and the elapsed waiting time, and is of the form . Coupling of this kind generates larger displacements for longer waiting times, however, decouples on longer timescales. Such coupling is proposed to be physically relevant to systems in which diffusion is driven by the development of internal stresses which release and develop cyclically. This article outlines the associated generalised diffusion equation (GDE), which describes the time evolution of the position PDF, . The solution for is obtained using the properties of the Fox H function, both in terms of its transform properties but also its expansion theorems. The second moment and the asymptotic features of the solution are extracted. The relaxation of back to the solution of a decoupled-type GDE is highlighted.

Advancing the knowledge of solute transport in the presence of bound water in mixed porous media based on low-field nuclear magnetic resonance

Bound water (BW) has significant effects on the physical and chemical properties of soils and other geologic formations. However, little is known about its effect on the non-Fickian transport of solute in porous media. This paper investigated the effect of BW on the non-Fickian solute transport characteristics in mixed porous media. Different porous media of quartz sand with variable contents of powdered minerals were prepared. The low-field nuclear magnetic resonance (LF-NMR) was used to quantitatively characterise BW and its spatial distribution. Then solute transport experiments were conducted in different prepared porous media and the measured breakthrough curves (BTCs) were analysed using the advection–dispersion equation (ADE), mobile-immobile (MIM) and continuous time random walk (CTRW) models. The results revealed that all measured BTCs exhibited the characteristics of early arrival and late-time tailing. However, stronger non-Fickian characteristics were observed as BW increased in content, indicating that BW influenced the non-Fickian transport of the solute. The optimised immobile water fraction of the MIM model consistently increased with increasing estimated BW fraction. Moreover, the coefficient of mass transfer rate (α) of MIM decreased as BW increased, meaning that the solute exchange between the mobile and immobile fluids became slower and consequently yielded strongly tailed BTCs. From the CTRW model, the solute transport velocity (vCTRW) increased with increasing BW, leading to a reduction in the breakthrough time of the solute. An increase in BW yielded a decreasing power law exponent (βCTRW) of the CTRW model, implying an enhancement of the non-Fickian transport characteristics. Furthermore, the porous media with BW<16.4% showed moderately non-Fickian solute transport whereas those with BW≥16.4% exhibited highly non-Fickian solute transport.

Discrete approximation to Brownian motion with varying dimension in bounded domains

In this paper, we study a discrete approximation to Brownian motion with varying dimension (BMVD) introduced by Chen and Lou in their 2019 paper with continuous time random walks on square lattices. The state space of BMVD contains a 2-dimensional component, a 3-dimensional component, and a “darning point” which joins these two components. Such a state space is equipped with the geodesic distance under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous-time reversible random walks with exponential holding times.

An application of a multiindex, time-fractional differential equation to evaluate heterogeneous, fractured rocks

A multiindex, distributed fractional differential equation is derived and solved in terms of the Laplace transformation. Potential applications of the proposed model include the study of fluid flow in heterogeneous rocks, the examination of bimodal fluid exchange between mobile-immobile regions in groundwater systems, the incorporation of the existence of liesegang bands in fractured rocks, and addressing the influences of faulted and other skin regions at interfaces, among others. Asymptotic solutions that reveal the structure of the resulting solutions are presented; in addition, they provide for ensuring the accuracy of the numerical computations. Fractional flux laws based on Continuous Time Random Walks (CTRW) serve as a linchpin to account for complex geological considerations that arise in the flow of fluids in heterogeneous rocks. Results are intended to be applied at the Theis scale when combined with geological/geophysical models and production statistics to all aspects of subsurface flow: production of geothermal and hydrocarbon fluids, injection of fluids into aquifers, geologic sequestration and hazardous waste disposal. Results may be extended to study the role of complex wellbores such as horizontal and fractured wells and more complex geological considerations such as faulted systems.

Almost Sure Limit Points for Continuous Time Random Walks

Here we obtain almost sure limit points for a properly normalized partial sum continuous time random walk, where a continuous time random walk means a random walk subordinated to a renewal process. Continuous time random walks are used in physics to model anomalous diffusion.

A class of anomalous diffusion epidemic models based on CTRW and distributed delay

In recent years, the epidemic model with anomalous diffusion has gained popularity in the literature. However, when introducing anomalous diffusion into epidemic models, they frequently lack physical explanation, in contrast to the traditional reaction–diffusion epidemic models. The point of this paper is to guarantee that anomalous diffusion systems on infectious disease spreading remain physically reasonable. Specifically, based on the continuous-time random walk (CTRW), starting from two stochastic processes of the waiting time and the step length, time-fractional space-fractional diffusion, time-fractional reaction–diffusion and fractional-order diffusion can all be naturally introduced into the SIR (S: susceptible, I: infectious and R: recovered) epidemic models, respectively. The three models mentioned above can also be applied to create SIR epidemic models with generalized distributed time delays. Distributed time delay systems can also be reduced to existing models, such as the standard SIR model, the fractional infectivity model and others, within the proper bounds. Meanwhile, as an application of the above stochastic modeling method, the physical meaning of anomalous diffusion is also considered by taking the SEIR (E: exposed) epidemic model as an example. Similar methods can be used to build other types of epidemic models, including SIVRS (V: vaccine), SIQRS (Q: quarantined) and others. Finally, this paper describes the transmission of infectious disease in space using the real data of COVID-19.

Fractional Calculus Extension of the Kinetic Theory of Fluids: Molecular Models of Transport within and between Phases

The application of fractional calculus in the field of kinetic theory begins with questions raised by Bernoulli, Clausius, and Maxwell about the motion of molecules in gases and liquids. Causality, locality, and determinism underly the early work, which led to the development of statistical mechanics by Boltzmann, Gibbs, Enskog, and Chapman. However, memory and nonlocality influence the future course of molecular interactions (e.g., persistence of velocity and inelastic collisions); hence, modifications to the thermodynamic equations of state, the non-equilibrium transport equations, and the dynamics of phase transitions are needed to explain experimental measurements. In these situations, the inclusion of space- and time-fractional derivatives within the context of the continuous time random walk (CTRW) model of diffusion encodes particle jumps and trapping. Thus, we anticipate using fractional calculus to extend the classical equations of diffusion. The solutions obtained illuminate the structure and dynamics of the materials (gases and liquids) at the molecular, mesoscopic, and macroscopic time/length scales. The development of these models requires building connections between kinetic theory, physical chemistry, and applied mathematics. In this paper, we focus on the kinetic theory of gases and liquids, with particular emphasis on descriptions of phase transitions, inter-phase mixing, and the transport of mass, momentum, and energy. As an example, we combine the pressure–temperature phase diagrams of simple molecules with the corresponding anomalous diffusion phase diagram of fractional calculus. The overlap suggests links between sub- and super-diffusion and molecular motion in the liquid and the vapor phases.

Exact calculation of the mean first-passage time of continuous-time random walks by nonhomogeneous Wiener–Hopf integral equations

We study the mean first-passage time (MFPT) for asymmetric continuous-time random walks in continuous-space characterised by waiting-times with finite mean and by jump-sizes with both finite mean and finite variance. In the asymptotic limit, this well-controlled process is governed by an advection-diffusion equation and the MFPT results to be finite when the advecting velocity is in the direction of the boundary. We derive a nonhomogeneous Wiener–Hopf integral equation that allows for the exact calculation of the MFPT by avoiding asymptotic limits and it emerges to depend on the whole distribution of the jump-sizes and on the mean-value only of the waiting-times, thus it holds for general non-Markovian random walks. Through the case study of a quite general family of asymmetric distributions of the jump-sizes that is exponential towards the boundary and arbitrary in the opposite direction, we show that the MFPT is indeed independent of the jump-sizes distribution in the opposite direction to the boundary. Moreover, we show also that there exists a length-scale, which depends only on the features of the distribution of jumps in the direction of the boundary, such that for starting points near the boundary the MFPT depends on the specific whole distribution of jump-sizes, in opposition to the universality emerging for starting points far-away from the boundary.

Subdiffusive continuous time random walks with power-law resetting

In the present work we revisit the problem of the behavior of a subdiffusive continuous time random walk (CTRW) under resetting. The resetting process is considered as a renewal process with power-law distribution of waiting times between the resetting events. We consider both the case of complete resetting, when an ordinary CTRW starts anew after the resetting event, and the case of incomplete resetting, when the internal memory of CTRW is nor erased by the resetting event, and the CTRW restarts as an aged one. Using a special representation for the waiting time distribution in resetting, we obtained closed-form expressions for the probability density of displacements and for the mean first passage time to a given point under complete resetting, and asymptotic forms for the probability density of displacements (including prefactors) in the case of incomplete resetting.

Anomalous transport in a porous medium with randomly packed ellipse cavities

We investigate the transport of nonreactive tracers in a binary porous medium with randomly packed ellipse fluid-filled cavities. Anomalous transport features, such as early arrival time and long tailing, are observed due to the high contrast in medium properties and highly complex structure of fluid velocity. We use a particle tracking method to quantify transport features of the domain. Then, a continuous time random walk (CTRW) framework builds on tracer transitions in time and space to represent an upscaled model. We study the effect of several key parameters on the anomalous transport process. The parameters include the cavity aspect ratio, porous background permeability, and the Peclet (Pe) number. With the increase in Pe, a longer tailing and a larger residence time are observed, which presents a stronger anomalous feature. A similar situation corresponds to decreased porous medium permeability, which results in wider breakthrough curves. A longer tailing arises in the case of more elongated cavity of larger aspect ratio. The purely advective transport in the medium is investigated at Pe = ∞. This is considered a limit case for the anomalous behavior of the system. One can refer to this case as the most extended tail possible for each cavity arrangement. The widest breakthrough curves for a Pe = ∞ correspond to larger aspect ratios of the cavity and a lower permeable matrix. We show that the upscaled CTRW model can closely predict breakthrough curves in a binary medium with randomly distributed ellipse cavities. These findings give new insight into transport in vesicular and vuggy porous media.

Modelling anomalous diffusion in semi-infinite disordered systems and porous media

For an effectively one-dimensional, semi-infinite disordered system connected to a reservoir of tracer particles kept at constant concentration, we provide the dynamics of the concentration profile. Technically, we start with the Montroll–Weiss equation of a continuous time random walk with a scale-free waiting time density. From this we pass to a formulation in terms of the fractional diffusion equation for the concentration profile in a semi-infinite space for the boundary condition , using a subordination approach. From this we deduce the tracer flux and the so-called breakthrough curve (BTC) at a given distance from the tracer source. In particular, BTCs are routinely measured in geophysical contexts but are also of interest in single-particle tracking experiments. For the ‘residual’ BTCs, given by , we demonstrate a long-time power-law behaviour that can be compared conveniently to experimental measurements. For completeness we also derive expressions for the moments in this constant-concentration boundary condition.

Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the ‘aging effect’ as a hallmark of non-Markovianity where the discrete-time version of the ‘aging renewal process’ comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a t2-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (‘narrow’ with finite mean and variance) the walk exhibits normal diffusion and for ‘broad’ waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.

Incorporating multiple magnetic resonance diffusion models to differentiate low- and high-grade adult gliomas: a machine learning approach.

Accurate grading of gliomas is a challenge in imaging diagnosis. This study aimed to evaluate the performance of a machine learning (ML) approach based on multiparametric diffusion-weighted imaging (DWI) in differentiating low- and high-grade adult gliomas. A model was developed from an initial cohort containing 74 patients with pathology-confirmed gliomas, who underwent 3 tesla (3T) diffusion magnetic resonance imaging (MRI) with 21 b values. In all, 112 histogram features were extracted from 16 parameters derived from seven diffusion models [monoexponential, intravoxel incoherent motion (IVIM), diffusion kurtosis imaging (DKI), fractional order calculus (FROC), continuous-time random walk (CTRW), stretched-exponential, and statistical]. Feature selection and model training were performed using five randomly permuted five-fold cross-validations. An internal test set (15 cases of the primary dataset) and an external cohort (n=55) imaged on a different scanner were used to validate the model. The diagnostic performance of the model was compared with that of a single DWI model and DWI radiomics using accuracy, sensitivity, specificity, and the area under the curve (AUC). Seven significant multiparametric DWI features (two from the stretched-exponential and FROC models, and three from the CTRW model) were selected to construct the model. The multiparametric DWI model achieved the highest AUC (0.84, versus 0.71 for the single DWI model, P<0.05), an accuracy of 0.80 in the internal test, and both AUC and accuracy of 0.76 in the external test. Our multiparametric DWI model differentiated low- (LGG) from high-grade glioma (HGG) with better generalization performance than the established single DWI model. This result suggests that the application of an ML approach with multiple DWI models is feasible for the preoperative grading of gliomas.

First passage statistics of active random walks on one and two dimensional lattices

We investigate the first passage statistics of active continuous time random walks with Poissonian waiting time distribution on a one dimensional infinite lattice and a two dimensional infinite square lattice. We study the small and large time properties of the probability of the first return to the origin as well as the probability of the first passage to an arbitrary lattice site. It is well known that the occupation probabilities of an active particle resemble that of an ordinary Brownian motion with an effective diffusion constant at large times. Interestingly, we demonstrate that even at the leading order, the first passage probabilities are not given by a simple effective diffusion constant. We demonstrate that at late times, activity enhances the probability of the first return to the origin and the probabilities of the first passage to lattice sites close enough to the origin, which we quantify in terms of the Péclet number. Additionally, we derive the first passage probabilities of a symmetric random walker and a biased random walker without activity as limiting cases. We verify our analytic results by performing kinetic Monte Carlo simulations of an active random walker in one and two dimensions.

Learning attack-defense response in continuous-time discrete-states Stackelberg Security Markov games

ABSTRACT Researchers have become interested in security games in recent decades as a result of its successful application in real-world security issues. The security model is based on the Stackelberg Security Game (SSG), in which defenders (leaders) select a defensive strategy based on the optimal reaction of attackers (followers), who, at equilibrium, select the predicted assaulting strategy as a response. These applications, on the other hand, do not account for the time constraints posed by the game’s players’ journey time. Furthermore, players should be able to cope with dynamic settings in which their knowledge of the environment changes on a regular basis, allowing them to perform more effectively. This research proposes a security model based on a continuous-time Reinforcement Learning (RL) approach implemented using a temporal difference method that takes prior information into account to address these issues. We use a controlled, ergodic continuous-time Markov game to model the SSG. The game framework model assumes that all information is available. We calculate the number of transitions over a time interval divided by the entire value of the holding time to estimate the transition rates. The arithmetic mean of the observed cost of the individual players is used to estimate the cost for defenders and attackers. An iterated proximal/gradient approach is used to calculate the SSG equilibrium point. We offer a continuous-time random walk method for game implementation. In a numerical case relevant to rain-forest hazards, we analyze the performance of the suggested RL security solution and discuss the problems that should be considered in future.

Assessment of Prognostic Factors and Molecular Subtypes of Breast Cancer With a Continuous-Time Random-Walk MR Diffusion Model: Using Whole Tumor Histogram Analysis.

The continuous-time random-walk (CTRW) diffusion model to evaluate breast cancer prognosis is rarely reported. To investigate the correlations between apparent diffusion coefficient (ADC) and CTRW-specific parameters with prognostic factors and molecular subtypes of breast cancer. Retrospective. One hundred fifty-seven women (median age, 50 years; range, 26-81 years) with histopathology-confirmed breast cancer. Simultaneous multi-slice readout-segmented echo-planar imaging at 3.0T. The histogram metrics of ADC, anomalous diffusion coefficient (D), temporal diffusion heterogeneity (α), and spatial diffusion heterogeneity (β) were calculated for whole-tumor volume. Associations between histogram metrics and prognostic factors (estrogen receptor [ER], progesterone receptor [PR], human epidermal growth factor receptor 2 [HER2], and Ki-67 proliferation index), axillary lymph node metastasis (ALNM), and tumor grade were assessed. The performance of histogram metrics, both alone and in combination, for differentiating molecular subtypes (HER2-positive, Luminal or triple negative) was also assessed. Comparisons were made using Mann-Whitney test between different prognostic factor statuses and molecular subtypes. Receiver operating characteristic curve analysis was used to assess the performance of mean and median histogram metrics in differentiating the molecular subtypes. A P value <0.05 was considered statistically significant. The histogram metrics of ADC, D, and α differed significantly between ER-positive and ER-negative status, and between PR-positive and PR-negative status. The histogram metrics of ADC, D, α, and β were also significantly different between the HER2-positive and HER2-negative subgroups,and between ALNM-positive and ALNM-negative subgroups. The histogram metrics of α and β significantly differed between high and low Ki-67 proliferation subgroups, and between histological grade subgroups. The combination of αmean and βmean achieved the highest performance (AUC=0.702) to discriminate the Luminal and HER2-positive subtypes. Whole-tumor histogram analysis of the CTRW model has potential to provide additional information on the prognosis and intrinsic subtyping classification of breast cancer. 4 TECHNICAL EFFICACY: Stage 2.