Framework Of Continuous Time Random Walk Research Articles

Diffusion in Porous Rock Is Anomalous.

Molecular diffusion of chemical species in subsurface environments─rock formations, soil sediments, marine, river, and lake sediments─plays a critical role in a variety of dynamic processes, many of which affect water chemistry. We investigate and demonstrate the occurrence of anomalous (non-Fickian) diffusion behavior, distinct from classically assumed Fickian diffusion. We measured molecular diffusion through a series of five chalk and dolomite rock samples over a period of about two months. We demonstrate that in all cases, diffusion behavior is significantly different than Fickian. We then analyze the results using a continuous time random walk framework that can describe anomalous diffusion in heterogeneous porous materials such as rock. This methodology shows extreme long-time tailing of tracer advance as compared to conventional Fickian diffusion processes. The finding that distinct anomalous diffusion occurs ubiquitously implies that diffusion-driven processes in subsurface zones should be analyzed using tools that account for non-Fickian diffusion.

Langevin picture of anomalous diffusion processes in expanding medium.

The expanding medium is very common in many different fields, such as biology and cosmology. It brings a nonnegligible influence on particle's diffusion, which is quite different from the effect of an external force field. The dynamic mechanism of a particle's motion in an expanding medium has only been investigated in the framework of a continuous-time random walk. To focus on more diffusion processes and physical observables, we build the Langevin picture of anomalous diffusion in an expanding medium, and conduct detailed analyses in the framework of the Langevin equation. With the help of a subordinator, both subdiffusion process and superdiffusion process in the expanding medium are discussed. We find that the expanding medium with different changing rate (exponential form and power-law form) leads to quite different diffusion phenomena. The particle's intrinsic diffusion behavior also plays an important role. Our detailed theoretical analyses and simulations present a panoramic view of investigating anomalous diffusion in an expanding medium under the framework of the Langevin equation.

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.

Analytical Investigations into Anomalous Diffusion Driven by Stress Redistribution Events: Consequences of Lévy Flights

This research is concerned with developing a generalised diffusion equation capable of describing diffusion processes driven by underlying stress-redistributing type events. The work utilises the development of an appropriate continuous time random walk framework as a foundation to consider a new generalised diffusion equation. While previous work has explored the resulting generalised diffusion equation for jump-timings motivated by stick-slip physics, here non-Gaussian probability distributions of the jump displacements are also considered, specifically Lévy flights. This work illuminates several features of the analytic solution to such a generalised diffusion equation using several known properties of the Fox H function. Specifically demonstrated are the temporal behaviour of the resulting position probability density function, and its normalisation. The reduction of the proposed form to expected known solutions upon the insertion of simplifying parameter values, as well as a demonstration of asymptotic behaviours, is undertaken to add confidence to the validity of this equation. This work describes the analytical solution of such a generalised diffusion equation for the first time, and additionally demonstrates the capacity of the Fox H function and its properties in solving and studying generalised Fokker–Planck equations.

Advective Trapping in the Flow Through Composite Heterogeneous Porous Media

We study the mechanisms of advective trapping in composite porous media that consist of circular inclusions of distributed hydraulic conductivity embedded in a high conductivity matrix. Advective trapping occurs when solute enters low velocity regions in the media. Transport is analyzed in terms of breakthrough curves measured at the outlet of the system. The curve’s peak behavior depends on the volume fraction occupied by the inclusions, while the tail behavior depends on the distribution of hydraulic conductivity values. In order to quantify the observed behaviors, we derive two equivalent upscaled transport models. First, we derive a Lagrangian trapping model using the continuous-time random walk framework that is parameterized in terms of volume fraction and the distribution of conductivities in the inclusions. Second, we establish a non-local partial differential equation for the mobile solute concentration by volume averaging of the microscale transport equation. We show the equivalence between the two models as well as (first-order) multirate mass transfer models. The upscaled approach parameterized by medium and flow properties captures all features of the observed solute breakthrough curves and sheds new light on the modeling of advective trapping in heterogeneous media.

Anomalous diffusion driven by the redistribution of internal stresses.

This article explores the mathematical description of anomalous diffusion, driven not by thermal fluctuations but by internal stresses. A continuous time random walk framework is outlined in which the waiting times between displacements (jumps), generated by the dynamics of internal stresses, are described by the generalized Γ distribution. The associated generalized diffusion equation is then identified. The solution to this equation is obtained as an integral over an infinite series of Fox H functions. The probability density function is identified as initially non-Gaussian, while at longer timescales Gaussianity is recovered. Likewise, the second moment displays a transient nature, shifting between subdiffusive and diffusive character. The potential application of this mathematical description to the quaking observed in several soft-matter systems is discussed briefly.

The chemical continuous time random walk framework for upscaling transport limitations in fluid–solid reactions

Fluid-solid reactions play a key role in a large range of biogeochemical processes. Transport limitations at the pore scale limit the amount of solute available for reaction, so that reaction rates measured under well-mixed conditions tend to strongly overestimate rates occurring in natural and engineered systems. Although different models have been proposed to capture this phenomenon, linking pore-scale structure, flow heterogeneity, and local reaction kinetics to upscaled effective kinetics remains a challenging problem. We present a new theoretical framework to quantify these dynamics based on the chemical continuous time random walk framework. We study a fluid–solid reaction with the fluid phase undergoing advective–diffusive transport. We consider a catalytic degradation reaction, AF+BS→BS, where AF is in fluid phase and BS is in solid phase and homogeneous over the fluid–solid interface, allowing us to focus on the role of transport limitations and medium structure. Our approach is based on the concept of inter-reaction times, which result from the times between contacts of transported reactants with the solid phase. We use this formulation to quantify the global kinetics of fluid-reactant mass and test our predictions against numerical simulations of advective–diffusive transport in stratified channel flow and Stokes flow through a beadpack. The theory captures the decrease of effective reaction rates compared to the well-mixed prediction with increasing Damköhler number due to transport limitations. Although we consider simple kinetics and media, these findings will contribute to the understanding and modeling of the effect of transport limitations in more complex reactive transport problems.

The Effect of varying correlation lengths on Anomalous Transport

Conventional concepts for transport in porous media assume that the heterogeneous distribution of hydraulic conductivities is the source for the contaminant temporal and spatial heavy tail. This tailing, known as anomalous or non-Fickian transport, can be captured by the β parameter in the continuous-time random walk framework. This study shows that with the increase in spatial correlation length between these heterogeneous distributions of hydraulic conductivities, the transport’s anomaly reduces; yet, the β value is unchanged, suggesting a topological component of the conductivity field, captured by the β. This finding is verified by an analysis of the solute transport, showing that the changing conductivity values have a moderate effect on the transport shape.

Nonlocal transport models for capturing solute transport in one‐dimensional sand columns: Model review, applicability, limitations and improvement

AbstractModelling pollutant transport in water is one of the core tasks of computational hydrology, and various physical models including especially the widely used nonlocal transport models have been developed and applied in the last three decades. No studies, however, have been conducted to systematically assess the applicability, limitations and improvement of these nonlocal transport models. To fill this knowledge gap, this study reviewed, tested and improved the state‐of‐the‐art nonlocal transport models, including their physical background, mathematical formula and especially the capability to quantify conservative tracers moving in one‐dimensional sand columns, which represents perhaps the simplest real‐world application. Applications showed that, surprisingly, neither the popular time‐nonlocal transport models (including the multi‐rate mass transfer model, the continuous time random walk framework and the time fractional advection‐dispersion equation), nor the spatiotemporally nonlocal transport model (ST‐fADE) can accurately fit passive tracers moving through a 15‐m‐long heterogeneous sand column documented in literature, if a constant dispersion coefficient or dispersivity is used. This is because pollutant transport in heterogeneous media can be scale‐dependent (represented by a dispersion coefficient or dispersivity increasing with spatiotemporal scales), non‐Fickian (where plume variance increases nonlinearly in time) and/or pre‐asymptotic (with transition between non‐Fickian and Fickian transport). These different properties cannot be simultaneously and accurately modelled by any of the transport models reviewed by this study. To bypass this limitation, five possible corrections were proposed, and two of them were tested successfully, including a time fractional and space Hausdorff fractal model which minimizes the scale‐dependency of the dispersion coefficient in the non‐Euclidean space, and a two‐region time fractional advection‐dispersion equation which accounts for the spatial mixing of solute particles from different mobile domains. Therefore, more efforts are still needed to accurately model transport in non‐ideal porous media, and the five model corrections proposed by this study may shed light on these indispensable modelling efforts.

Characterization and upscaling of hydrodynamic transport in heterogeneous dual porosity media

We study the upscaling of pore-scale transport of passive solute in a carbonate rock sample. It is characterized by microporous regions displaying heterogeneous porosity distribution that are accessible due to diffusion only, and a strongly heterogeneous mobile pore space, characterized by a broad distribution of flow velocities. We observe breakthrough curves that are characterized by strong tailing, which can be attributed to velocity variability in the flowing medium portion, and solute retention in the microporous space. Using accurate numerical flow and transport simulations, we separate these two mechanisms by analyzing the statistics of residence times in the mobile phase, and the trapping and residence time statistics in the mmobile phase. We employ a continuous time random walk framework in order to upscale transport using a particle based implementation of mobile-immobile mass transfer, and heterogeneous advection. This approach is based on the statistics of the characteristic mobile and immobile residence times, and mass transfer rates between the two continua. While classical mobile-immobile approaches model mass transfer as a constant rate process, we find that the trapping rate increases with increasing mobile residence times until it reaches a constant asymptotic value. Based on these findings and the statistical characteristics of travel and retention times, we derive an upscaled Lagrangian transport model that separates the processes of heterogeneous advection and diffusion in the immobile microporous space, and provides accurate descriptions of the observed non-Fickian breakthrough curves. These results shed light on transport upscaling in highly complex dual-porosity rocks for which mobile-immobile mass transfer are controlled by a dual multirate process controlled by the heterogeneity of both the flow field in the connected porosity and the diffusion in the no-flow regions.

Structural and temporal heterogeneities on networks

A heterogeneous continuous time random walk is an analytical formalism for studying and modeling diffusion processes in heterogeneous structures on microscopic and macroscopic scales. In this paper we study both analytically and numerically the effects of structural and temporal heterogeneities onto the diffusive dynamics on different types of networks. For this purpose we investigate how the distribution of the first passage time is affected by the global topological network properties and heterogeneities in the distributions of the travel times. In particular, we analyze transport properties of random networks and define network measures based on the first-passage characteristics. The heterogeneous continuous time random walk framework, presented in the paper, has potential applications in biology, social and urban science, search of optimal transport properties, analysis of the effects of heterogeneities or bursts in transportation networks.

The Role of Fractional Time-Derivative Operators on Anomalous Diffusion

The generalized diffusion equations with fractional order derivatives have shown be quite efficient to describe the diffusion in complex systems, with the advantage of producing exact expressions for the underlying diffusive properties. Recently, researchers have proposed different fractional-time operators (namely: the Caputo-Fabrizio and Atangana-Baleanu) which, differently from the well-known Riemann-Liouville operator, are defined by non-singular memory kernels. Here we proposed to use these new operators to generalize the usual diffusion equation. By analyzing the corresponding fractional diffusion equations within the continuous time random walk framework, we obtained waiting time distributions characterized by exponential, stretched exponential, and power-law functions, as well as a crossover between two behaviors. For the mean square displacement, we found crossovers between usual and confined diffusion, and between usual and sub-diffusion. We obtained the exact expressions for the probability distributions, where non-Gaussian and stationary distributions emerged. This former feature is remarkable because the fractional diffusion equation is solved without external forces and subjected to the free diffusion boundary conditions. We have further shown that these new fractional diffusion equations are related to diffusive processes with stochastic resetting, and to fractional diffusion equations with derivatives of distributed order. Thus, our results show that these new operators are a simple and efficient way for incorporating different structural aspects into the system, opening new possibilities for modeling and investigating anomalous diffusive processes.

Continuous time random walks for non-local radial solute transport

This study formulates and analyzes continuous time random walk (CTRW) models in radial flow geometries for the quantification of non-local solute transport induced by heterogeneous flow distributions and by mobile–immobile mass transfer processes. To this end we derive a general CTRW framework in radial coordinates starting from the random walk equations for radial particle positions and times. The particle density, or solute concentration is governed by a non-local radial advection–dispersion equation (ADE). Unlike in CTRWs for uniform flow scenarios, particle transition times here depend on the radial particle position, which renders the CTRW non-stationary. As a consequence, the memory kernel characterizing the non-local ADE, is radially dependent. Based on this general formulation, we derive radial CTRW implementations that (i) emulate non-local radial transport due to heterogeneous advection, (ii) model multirate mass transfer (MRMT) between mobile and immobile continua, and (iii) quantify both heterogeneous advection in a mobile region and mass transfer between mobile and immobile regions. The expected solute breakthrough behavior is studied using numerical random walk particle tracking simulations. This behavior is analyzed by explicit analytical expressions for the asymptotic solute breakthrough curves. We observe clear power-law tails of the solute breakthrough for broad (power-law) distributions of particle transit times (heterogeneous advection) and particle trapping times (MRMT model). The combined model displays two distinct time regimes. An intermediate regime, in which the solute breakthrough is dominated by the particle transit times in the mobile zones, and a late time regime that is governed by the distribution of particle trapping times in immobile zones. These radial CTRW formulations allow for the identification of heterogeneous advection and mobile-immobile processes as drivers of anomalous transport, under conditions relevant for field tracer tests.

Weak Ergodicity Breaking of Membrane Receptor Motion Stemming from Random Diffusivity

The application of techniques such as fluorescence correlation spectroscopy and single particle tracking to the plasma membrane of living cells continuously evidences subdiffusive motion of proteins and lipids. This anomalous motion is generally associated to the interplay of molecular crowding, diffusion barriers and specific interactions. Addressing the cause of subdiffusion is essential for understanding molecular mechanisms underlying cellular function, such as target search, kinetics of transport-limited reactions, trafficking and signalling. Recently, the subdiffusive motion of some cellular and membrane components has been associated to weak ergodicity breaking and attributed to transient immobilization caused by interactions with microtubules or actin cytoskeleton. These works have opened new questions about the role of nonergodic subdiffusion in the dynamics of living systems.By means of single particle tracking experiments, we show that the dynamics of DC-SIGN, a transmembrane receptor with unique pathogen recognition capabilities, also reveals subdiffusion with signatures of weak ergodicity breaking and aging. In contrast to other biological systems, we demonstrate that DC-SIGN nonergodic subdiffusion cannot be explained by transient immobilization, but rather is compatible with dynamical heterogeneity induced by spatiotemporal changes of diffusivity. While nonergodic diffusion due to long-lived immobilization events can be understood within the framework of continuous-time random walk, our experimental data are accurately interpreted through a new theoretical model describing anomalous transport in biological systems and complex media. A comprehensive analysis of mutated forms of the receptor allowed us to establish a link between receptor molecular structure, nonergodic diffusion and function. Our results underscore the role of spatiotemporal disorder in the dynamics of cell membrane receptors. In addition, they have broad implications to other heterogeneous systems where the occurrence of nonergodicity remains unexplored.

Structural disorder and anomalous diffusion in random packing of spheres.

Nowadays Nuclear Magnetic Resonance diffusion (dNMR) measurements of water molecules in heterogeneous systems have broad applications in material science, biophysics and medicine. Up to now, microstructural rearrangement in media has been experimentally investigated by studying the diffusion coefficient (D(t)) behavior in the tortuosity limit. However, this method is not able to describe structural disorder and transitions in complex systems. Here we show that, according to the continuous time random walk framework, the dNMR measurable parameter α, quantifying the anomalous regime of D(t), provides a quantitative characterization of structural disorder and structural transition in heterogeneous systems. To demonstrate this, we compare α measurements obtained in random packed monodisperse micro-spheres with Molecular Dynamics simulations of disordered porous media and 3D Monte Carlo simulation of particles diffusion in these kind of systems. Experimental results agree well with simulations that correlate the most used parameters and functions characterizing the disorder in porous media.

Predicting Intraday Price Distributions at High Frequencies

We propose extensions to the continuous-time random walk (CTRW) framework so far mainly developed within the econophysics community. Using numerical methods, we extend the CTRW framework with more general marginal distributions than previously proposed. There are two main findings in this respect: First, modeling the returns with a Student-t distribution gives at least as good price distribution predictions as the other previously proposed distributions in this framework. Second, a mixed-Weibull waiting time distribution fits exceptionally well to our three-week long Nasdaq OMX data. When combined with standard financial econometric GARCH and ACD models and intraday seasonality filtering procedures new to the CTRW framework, our models deliver realistic intraday Value-at-Risk and Expected Shortfall predictions. Compared to the basic CTRW model, the total average performance improvement is typically around 70 percent. The effect of filtering out memory in waiting times is particularly noticeable: around 50 percent of the total performance improvement. Overall, our extensions and filtering methods make the CTRW framework a useful tool for intraday risk management (trading), especially at high frequencies of less than a minute or so.

Experimental and modeling analysis of coupled non-Fickian transport and sorption in natural soils

We present experimental breakthrough curve (BTC) data and a modeling investigation of conservative and sorbing tracer transport in natural soils. By analyzing the data using the continuous time random walk (CTRW) model, we probe the traditional approach of using conservative tracer model parameters as a basis for quantifying the transport of sorbing solutes in the same domain when non-Fickian transport is present. Many known contaminants in groundwater are sorbed to the host solid porous medium, to varying extents, while being transported; this enhances the long tailing of BTCs which often already occurs because of the inherent non-Fickian nature of the transport. The CTRW framework has been shown to account very well for non-Fickian conservative (nonsorbing) transport. Here, we examine two BTC data sets in laboratory columns packed with natural soils; the first (previously analyzed by Mao and Ren (2004)) comprises transport of (conservative) bromide and (sorbing) atrazine tracers, while the second presents new data with bromide and tribromoneopentyl alcohol (TBNPA), a key flame retardant, as a sorbing solute. TBNPA has received little attention in the past, and is shown to be sorbed onto Bet Dagan soil in a nonlinear manner. We find that the transport behavior of bromide is non-Fickian in all cases, which is caused by the heterogeneity of the soil. Comparative model analysis of the non-Fickian BTCs of the conservative, and sorbing tracers and examination of the fitting parameters, exemplify the coupling between transport and adsorption/desorption processes. The difference in transport parameters used to match the conservative and sorbing data sets shows that conservative tracer parameters (average velocity and dispersion coefficient) are not valid for the transport of reactive tracers.

Distribution of Parental Genome Blocks in Recombinant Inbred Lines

We consider recombinant inbred lines obtained by crossing two given homozygous parents and then applying multiple generations of self-crossings or full-sib matings. The chromosomal content of any such line forms a mosaic of blocks, each alternatively inherited identically by descent from one of the parents. Quantifying the statistical properties of such mosaic genomes has remained an open challenge for many years. Here, we solve this problem by taking a continuous chromosome picture and assuming crossovers to be noninterfering. Using a continuous-time random walk framework and Markov chain theory, we determine the statistical properties of these identical-by-descent blocks. We find that successive block lengths are only very slightly correlated. Furthermore, the blocks on the ends of chromosomes are larger on average than the others, a feature understandable from the nonexponential distribution of block lengths.

Dissolution and precipitation dynamics during dedolomitization

We simulate the processes of dedolomitization and calcium carbonate precipitation using particle tracking. The study is stimulated by the results of a laboratory experiment that examined reactive transport of injected CaCl2/HCl, into a column of sucrosic dolomite particles, with a constant flow field. The injected fluid supplies Ca2+ and H+. Dedolomitization is a protonation reaction yielding carbonic acid; a subsequent deprotonation reaction yields , and reaction with the abundant Ca2+ forms the precipitate CaCO3. The dedolomitization and precipitation processes involve multistep, multispecies chemical reactions, with both irreversible and reversible stages. The particle tracking is governed by spatial and temporal distributions within a continuous time random walk framework. This accounts for the effects of disorder of heterogeneous media (leading to non‐Fickian transport) and includes the option of treating purely advective‐dispersive (Fickian) transport. The dynamics of dedolomitization are examined for different flow conditions and reaction rates. The fluctuations in the local velocity distributions, due to porosity changes, create conditions for positive feedbacks leading to development of preferential pathways, large‐scale nonlinearity, and precipitation banding. These features have been observed in the laboratory experiments and are now accounted for by the simulation results at similar time frames, velocities, and pH levels.

Spatio-temporal anomalous diffusion in heterogeneous media by nuclear magnetic resonance

In this paper, we describe nuclear magnetic resonance measurements of water diffusion in highly confined and heterogeneous colloidal systems using an anomalous diffusion model. For the first time, temporal and spatial fractional exponents, α and μ, introduced within the framework of continuous time random walk, are simultaneously measured by pulsed gradient spin-echo NMR technique in samples of micro-beads dispersed in aqueous solution. In order to mimic media with low and high level of disorder, mono-dispersed and poly-dispersed samples are used. We find that the exponent α depends on the disorder degree of the system. Conversely, the exponent μ depends on both bead sizes and magnetic susceptibility differences within samples. The new procedure proposed here may be a useful tool to probe porous materials and microstructural features of biological tissue.