Nonlocal Transport Model Research Articles

Vertical matrix advection dominates transient anomalous diffusion within fracture-matrix systems using a modified diffusion model

Pollutant transport in fractured media has been well analyzed using process-based diffusion models and concise, time-nonlocal models like the tempered fractional derivative (TFD) model. While traditional diffusion models are limited to specific transport processes, TFD model parameters lack hydrogeological meaning. In addressing these knowledge gaps, a modified diffusion model is proposed to augment the traditional diffusion model and offer hydrogeological interpretations for TFD parameters. This modified diffusion model incorporates transverse advection in low-permeability matrices, allowing for the quantification of transient anomalous diffusion in a single fracture-matrix system. This model offers a semi-analytical solution, systematically validated for its applicability. Memory function analysis establishes a quantitative connection between the modified diffusion model and the TFD model, uncovering a pivotal discovery in this study. Parameter sensitivity analysis further shows that the matrix characteristic rate defined as a function of (Vm)2/(4Dm), where Vm is the transverse matrix velocity and Dm is the transverse matrix diffusion coefficient, controls the temporal evolution of anomalous sub-diffusion in the fracture. Applications are performed to check the feasibility of the modified diffusion model. The developed modified diffusion model sheds light on quantifying complex transport in fracture-matrix systems and enhances the predictability of nonlocal transport models.

Space-time dependent non-local thermal transport effects on laser ablation dynamics in inertial confinement fusion

In inertial confinement fusion (ICF), electron thermal transport plays a key role in laser ablation and the subsequent implosion processes, which always exhibits intractable non-local effects. Simple modifications of the local Spitzer–Härm model with either an artificially-assumed constant flux limiter or a purely time-dependent one are applied to explain some experimental data, but fail to simultaneously reproduce the space-time evolution of the whole laser ablation process. Here, by carrying out a series of one-dimensional and two-dimensional radiation hydrodynamic simulations where the space-time-dependent non-local thermal transport model proposed by Schurt, Nicolaï and Busquet (the SNB model) are self-consistently included, we systematically study the non-local effects on the whole laser ablation dynamics including those occurring at the critical surface, the conduction zone and the ablation front. Different from those obtained previously, our results show that due to the non-local heat flow redistribution and redirection, at the critical surface the thermal flux is more inhibited, in the conduction zone the lateral thermal transport is suppressed, and ahead of the ablation front the plasma is preheated. When combined together they eventually result in significant improvement of the laser absorption efficiency, extension of the conduction zone, increase of both the mass ablation rate and shock velocity. Furthermore, the dependence of these laser ablation dynamics on different drive laser intensities is investigated, which provides beneficial enlightenments on potential laser pulse shaping and/or ignition scheme optimization in ICF.

Patterning of nonlocal transport models in biology: The impact of spatial dimension

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction–diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with ‘run-and-chase’ interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells.

Effect of non-local transport of hot electrons on the laser-target ablation

The non-local heat transport of hot electrons during high-intensity laser interaction with plasmas can preheat the fuel and limit the heat flow in inertial confinement fusion. It increases the entropy of the fuel and decreases the final compression. In this paper, the non-local electron transport model that is based on the improved Schurtz–Nicolaï–Busquet (SNB) algorithm has been embedded into the radiation hydrodynamic code and is benchmarked with two classical non-local transport cases. Then, we studied a 2ω laser ablating a CH target by using the non-local module. It is found that the non-local effect becomes significant when the laser intensity is above 1×1014W/cm2. The mass ablation rate from the SNB model is increased compared to that of the flux-limited model due to the lower coronal plasma temperature. This non-local model has a better agreement with the experimental results compared to that of the flux-limited model. The non-local transport is strongly dependent on the laser frequency, and the thresholds that the non-local transport should be considered are obtained for lasers of different frequencies. The appropriate flux-limiters that should be employed in the flux-limited model for different lasers are also presented. The results here should have a good reference for the laser-target ablation applications.

Quantifying nonlocal bedload transport: A regional-based nonlocal model for bedload transport from local to global scales

Recent studies have emphasized the importance of nonlocal models in characterizing bedload transport in natural rivers, particularly in mixed-size gravel beds or steep hillslopes. Nonlocality denotes that a quantity (flux) at a specific location x is dependent on the conditions in the surrounding area, as opposed to solely at the location itself. This concept applies in bedload transport even in planar flumes, where particles are entrained at an upstream position and travel a finite distance, ultimately contributing nonlocally to the sediment flux. However, existing bedload transport models, such as the advection–diffusion equation (ADE) or the fractional derivative equation (FDE) models, are inadequate in characterizing the nonlocal transport behavior of bedload at a regional scale. Large errors may arise from the lack of an accurate description of the nonlocal bedload transport processes at regional scales. This study proposes a regional-based nonlocal bedload transport model, which is conceptualized from the probabilistic Exner-based equations and the peridynamic (PD) differential operator. The PD model encapsulates the nonlocal motion of bedload sediments on the basis of the PD differential operator, by utilizing a pre-defined weight function and influence domain. Comparisons demonstrate that the PD model serves as a generalized tool connecting the local and the global models with different PD functions and influence domains. Its variability on kernel function and influence domain, enable it conveniently describe sub-, super-, and normal diffusion behaviors of bedload transport.

Models of thermal conduction and non-local transport of relevance to space physics with insights from laser–plasma theory

Models of solar and space plasmas require an accurate model for thermal transport. The simplest such model is to assume that the fluid approach is valid and that local transport models can be used. These local transport coefficients are derived under the assumption that the electron mean-free path is “small” compared to the temperature scale length. When this approximation breaks down, non-local transport models or thermal flux limiters must be used to maintain a physically realistic model. This article will review the background theory of how small is “small” for the mean-free path and what options there are for including non-local transport within the fluid framework. Much of this recent work has been motivated by laser–plasma theory, where mean-free paths can be large and the Spitzer–Harm approach is never used.

The nonlocal electron heat transport under the non-Maxwellian distribution in laser plasmas and its influence on laser ablation

The electron heat transport plays an important role in laser driven inertial confinement fusion. For the plasmas created by intense laser, the traditional Spitzer–Härm theory cannot accurately describe the electron heat transport process mainly due to two physical effects. First, the electron distribution function would significantly differ from the Maxwellian distribution because of the inverse bremsstrahlung heating. Second, the long mean free paths of heat carrying electrons relative to the temperature scale length indicate that the electron heat flux has the nonlocal feature. In 2020, we have developed a nonlocal electron heat transport model based on the non-Maxwellian electron distribution function (NM-NL model) to describe the electron heat flux in laser plasmas. Recently, this model is successfully incorporated into our radiation hydrodynamical code RDMG. In this article, we numerically investigated the electron heat flux in laser plasmas, especially the nonlocal feature of heat flux and the influence of the non-Maxwellian distribution. The influence of electron heat transport on laser ablation is also discussed. The simulated plasma conditions based on different electron heat transport models are presented and compared with experiments. Our results show that the nonlocal feature of heat flux and the influence of non-Maxwellian distribution function are considerable in plasmas heated by intense lasers.

A nonlocal electron transport model in the diffusion scaling of hydrodynamics

We analyze a model of nonlocal electron transport named P1-diffusion based on a spherical harmonic expansion in velocity space and a diffusion scaling, which makes it compatible with assumptions from magneto-hydrodynamics (MHD). An iterative, fully implicit (CFL-free, as defined by the Courant Friedrich Levy condition) and asymptotic preserving discretization is proposed, which necessitates the inversion of a possibly large number of—but small—linear systems. It is found accurate with respect to reference solutions from a Vlasov–Fokker–Planck–Maxwell code (based on a Polynomial expansion of order N, or PN expansion) on a series of tests, which are representative of the conduction zone in laser-created plasmas. Thereby, the present approach is a good candidate for being embedded in multi-D MHD codes.

Machine learning of nonlocal micro-structural defect evolutions in crystalline materials

The presence and evolution of defects that appear in the manufacturing process play a vital role in the failure mechanisms of engineering materials. In particular, the collective behavior of dislocation dynamics at the mesoscale leads to avalanche, strain bursts, intermittent energy spikes, and nonlocal interactions producing anomalous features across different time- and length-scales, directly affecting plasticity, void and crack nucleation. Discrete Dislocation Dynamics (DDD) simulations are often used at the meso-level, but the cost and complexity increase dramatically with simulation time. To further understand how the anomalous features propagate to the continuum, we develop a probabilistic model for dislocation motion constructed from the position statistics obtained from DDD simulations. We obtain the continuous limit of discrete dislocation dynamics through a Probability Density Function for the dislocation motion, and propose a nonlocal transport model for the PDF. We develop a machine-learning framework to learn the parameters of the nonlocal operator with a power-law kernel, connecting the anomalous nature of DDD to the origin of its corresponding nonlocal operator at the continuum, facilitating the integration of dislocation dynamics into multi-scale, long-time material failure simulations.

Global well-posedness, regularity and blow-up for the β-CCF model

The paper is concerned with the β-CCF model which is a 1D nonlocal nonlinear transport model. The velocity is coupled via a composition of the Hilbert transform and Riesz potentials, giving rise to three basic cases that reflect the balance between the nonlinearity and dissipation, namely the subcritical, critical and supercritical ranges. In our analysis we consider those three cases and obtain a set of results containing local well-posedness, global smoothness, eventual regularity and blow-up of solutions.

Progress in ultrafast spintronics research

Ultrafast spintronics is one of the most promising fields for information processing technology innovation. When compared with traditional electronic devices, new spin-based devices have advantages such as low power consumption, high speed, and nonvolatility. Exploring novel physical phenomena and mechanisms relating to electronic spins significantly promotes the development of next-generation spintronic devices. In this review, we outline a sequence of advances in the fields of ultrafast magnetics and spintronics. First, a quick overview of spintronics and the discovery of ultrafast demagnetization induced by a femtosecond laser is provided. The proposed physical mechanisms and theories of ultrafast spin dynamics are then summarized, including the phenomenological three-temperature model, computations based on the time-dependent density functional theory, the Elliott-Yafet spin-flip mechanism, and the nonlocal superdiffusive spin transport model. Following that, the most recent experimental and theoretical studies on all-optical switching of ferrimagnetic and ferromagnetic metals are thoroughly reviewed. Furthermore, the discovery of hot-electron spin transport induced by ultrashort lasers as well as its influence on ultrafast magnetization dynamics, ultrafast spin transfer torque, spin injection into semiconductors, and spin accumulation and dissipation in hot-electron transport are described. The advancements in the optimization and manipulation of spintronics-based terahertz emitters are reviewed and analyzed. Finally, we discuss the current challenges in ultrafast spintronics research as well as possible future studies.

A Dual Heterogeneous Domain Model for Upscaling Anomalous Transport With Multi‐Peaks in Heterogeneous Aquifers

AbstractNatural aquifers are often characterized by multiscale heterogeneity and complex flow networks, challenging the reliable simulation and prediction of contaminant transport. Classic stochastic transport models usually assume independent, identically distributed probability density functions when upscaling solute dynamics, but this assumption is not valid for media with multiscale heterogeneity. To address this issue, a dual heterogeneous domain model (DHDM) was proposed to quantify solute transport in heterogeneous aquifers where solute breakthrough curves (BTCs) exhibit multiple peaks and transient tailing behaviors. To efficiently solve the resultant DHDM, a Lagrangian solver was developed by combining the renewal‐reward process (to capture solute dynamics in each domain) and the state transition probability (to capture solute particle transfer between the two heterogeneous domains). The DHDM and its Lagrangian solver were then applied to simulate solute transport in three different aquifer settings (alluvial, fractured, and karst). Analyses showed that in the alluvial aquifer, early arrivals and bi‐peaks of the observed solute BTC might be caused by interconnected preferential flow paths consisting of high conductivity sediments, while the front edge or long tails of local peaks might be a result of small‐scale heterogeneity. Fracture networks and karst conduits also resulted in anomalous transport with multiple peaks, and mass transfer between the preferential flow channels and matrix led to solute retention. All these complex, coexisting anomalous transport characteristics can be simultaneously and accurately captured by the unifying DHDM approach, significantly expanding the capability of nonlocal transport models.

High order entropy stable and positivity-preserving discontinuous Galerkin method for the nonlocal electron heat transport model

The nonlocal electron heat transport model in laser heated plasmas plays a crucial role in inertial confinement fusion (ICF), and it is important to solve it numerically in an accurate and robust way. In this paper, we first develop an one-dimensional high-order entropy stable discontinuous Galerkin method for the nonlocal electron heat transport model. We further design our DG scheme to have the positivity-preserving property by using the scaling positivity-preserving limiter, which is shown, by a computer-aided proof, to have no extra time step constraint than that required by L2 stability. Next, we extend our one-dimensional scheme to two-dimensions on rectangular meshes and tensor product polynomial spaces. Numerical examples are given to verify the high-order accuracy and positivity-preserving properties of our scheme. By comparing the local and nonlocal electron heat transport models, we also observe more physical phenomena such as the flux reduction and the preheat effect from the nonlocal model.

Characterization of thermal transport and evolution of Au plasma in ICF experiments by Thomson scattering

This paper demonstrates the capability of optical Thomson scattering (OTS) to measure thermal transport, and provides support to radiation hydrodynamic and kinetic simulations of electron thermal transport and plasma evolution. OTS theory and plasma simulations are applied to the interpretation of experimental measurements of laser-produced plasma from spherical gold targets on the OMEGA laser facility. The dynamical form factor, S(k→,ω), of electron density fluctuations that is used in the fitting of Thomson scattering spectra includes ion–ion collisions and effects of non-Maxwellian distribution functions. OTS measurements and their interpretation are consistent with the nonlocal transport model in radiation hydrodynamic simulations as well as with kinetic simulations in the second half of the probe pulse duration. In particular, the reversal of heat transport during cooling is observed to be consistent with simulations, while some discrepancies are noted during the initial heating of the Au targets.

Atoms and Photons: Kinetic Equations with Delay

In his recent works, the author drew attention to the fact that the extraction of a part of a closed Hamiltonian system turns the original well-known Liouville differential equation into an integro-differential equation with a delayed time argument, which describes the dynamics of the selected subsystem as an open system. It was shown that the integral operator can be represented in the form of a fractional differential operator of distributed order. In this paper, we show how the kinetic theory of the system “atoms+photons” is transformed for the subsystem consisting of excited atoms. We present the derivation of the telegraph equation with delay, derive the Biberman–Holstein equation in the fractional differential form (with the fractional Laplace operator), and discuss boundary effects in the nonlocal transport model. The final section is devoted to laser technologies, which include free-electron lasers and laser cooling of atoms.

A unified non-local fluid transport model for heterogeneous saturated porous media

A unified non-local fluid transport model is proposed for heterogeneous saturated porous media. First, a non-local governing equation of fluid transport is presented based on the Non-ordinary State-based Peridynamics (NOSB-PD), and a novel flux vector state and a non-local Darcy’s law is proposed to illustrate non-local fluid transport effects in heterogeneous saturated porous media. The proposed flux state vector ensures the non-local governing equation of fluid transport model in heterogeneous saturated porous media degenerates into the local one when the horizon of each material point approaches to zero. A non-local pressure gradient at each material point can be obtained by averaging all the flux state vectors in its horizon, which unifies the weak and strong discontinuities of pressure in heterogeneous saturated porous media in a consistent way. Second, a variational formulation of the unified non-local fluid transport model is developed. This variational formulation enables the proposed model to deal with complex boundary conditions, which is difficult or even impossible for current fluid transport models based on Bond-based Peridynamics (BB-PD) and Ordinary State-based Peridynamics (OSB-PD). Third, a fully implicit algorithm combined with the Newton–Raphson method is introduced to solve the general non-linear problems in the heterogeneous saturated porous media. The penalty function method is employed to eliminate the zero-energy mode oscillation in NOSB-PD. Further, numerical examples demonstrate that the proposed model is accurate and can well capture the weak and strong discontinuities of pressure at the material interfaces and crack surfaces in the heterogeneous saturated porous media. Numerical examples also indicate that the model can effectively eliminate the zero-energy mode oscillation inherently rooted in the NOSB-PD, and thus reaches a stable numerical solution.

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.

A distributed-order time fractional derivative model for simulating bimodal sub-diffusion in heterogeneous media

Bimodal transport due to dual-mobile advection or mass exchange between mobile and immobile zones has been widely observed for pollutants moving in heterogeneous media. Existing nonlocal transport models, however, cannot capture either heterogeneity in mobile domains or bi-peak concentration phenomena for breakthrough curves (BTCs) of solute transport in complex media. Therefore, this study proposed a dual heterogeneous domain model (DHDM) framework, conceptualized from the distributed-order time fractional derivative that incorporates a broader spectrum of particle movement in the dual-domain system than the classical single-domain model. A Lagrangian scheme was developed to solve the DHDM using the Bernoulli trial with the transition probability simulating mass exchange between different domains. Phase transition probabilities calculated based on the zeroth spatial moments of dual domain transport equations were used to simulate the particle partitioning between different domains. Applications to transport experiments in silt–clay columns conducted in our laboratory showed that the DHDM yielded significant improvement for simulating solute transport behavior compared to the single-domain anomalous transport model. Parameter analysis of the DHDM further revealed that the velocity disparity of the two domains controlled the BTC rising time and the shape of the second peak in the BTC. When the velocity of the slow domain is extremely small, the second BTC peak develops a ‘shoulder’ characteristic during the late-time tail of the BTC. This study improved our understanding for bimodal transport in a dual domain system and provides a feasible tool for capturing a wide range of bimodal transport in complex media.

A non-local heat transport model in solids with discontinuities via Lagrangian particle method

A new Lagrangian non-local diffusion model, designed as a meshless particle simulation method, is proposed to predict the thermal response of solids involving discontinuous. The main idea is to understand the heat transfer process of solids via a Lagrangian treatment of a particle-discretized system, which provides a more general physical representation of the heat transfer process of solids. In contrast to the traditional differential model, the proposed mathematical model is expressed via an integral form, which can easily handle the case involving discontinuities. The spatial convergence studies show that the proposed model converges to its associated non-local solution while increasing the number of particles in a fixed cut-off radius; and it can reach the local exact solution when the cut-off radius is close to zero. The numerical examples not only verify that the proposed diffusion model converges to the continuum heat conduction model, but also show the capability of its application to heat transfer problems involving discontinuities. In particular, the computational performance of the proposed non-local thermal model is also demonstrated its computational performance in the case with practical crack propagation in a two-dimensional plate. In addition, the specific comparison between the proposed method and the well-developed peridynamics approach for heat conduction problems is carefully described with rigor via mathematical proofs and numerical results. Specifically, the proposed model is shown to be a variation of the corresponding peridynamic-type thermal diffusion model. This model not only has a formulation consistent with that of peridynamics in solid mechanics but also shows a better performance in the case with sharp corners than that of the peridynamic diffusion in [1].

Upscaling non-linear reactive transport in correlated velocity fields

While commonly non-local transport models have been shown to reproduce breakthrough curves resulting from transport in heterogeneous aquifers successfully, open questions include the formal link between the upscaled governing equations and the sub-scale heterogeneity, and the ability to account for the effect of heterogeneity on effective chemical reaction rates in the presence of non-linear multi-component reactions. Time-domain random walk approaches based on velocity Markov models provide a framework to resolve these issues by incorporating the spatial correlation of velocity of a solute particle in consecutive locations along its trajectory. These approaches often rely on particle tracking approaches, which, however, can be computationally burdensome especially when non-linear reactions are sought to be modeled. In this paper, an integro-differential equation is proposed for upscaling multi-component reactive transport in heterogeneous media that relies on copulas for representing the velocity correlation structure. For this purpose, we express concentration or flux of solutes as a distribution over their velocity. We then derive the integro-differential equation that governs the evolution of concentration distribution over a quantity defined as velocity-rank. In this way, the spatial evolution of breakthrough curves away from the source is predicted based on ergodic cross-sectional velocity distributions and a parameterized copula function which expresses the correlation between velocity-ranks of a solute particle along its trajectory. We demonstrate the validity of the proposed model by comparing breakthrough curves for conservative and non-linearly reacting solutes based on realizations of hydraulic conductivity fields to the results of the upscaled model.