Continuous Time Random Walk Research Articles

An exactly solvable model for non-Fickian transport in dynamically heterogeneous media

Abstract Diffusion, observed in various condensed phases, finds its theoretical background in Einstein’s theory of Brownian motion, characterized by the linear time-dependence of mean square displacement (MSD) denoting Fickian behavior and the Gaussian distribution of particle displacement. Nevertheless, diverse systems exhibit either non-linear, non-Fickian time-dependence of the MSD or non-Gaussian displacement distribution. Montroll and Weiss’s continuous-time random walk (CTRW) model and the stochastic diffusivity (SD) model have provided insights into anomalous diffusion phenomena and Fickian-yet-non-Gaussian transport in dynamically heterogeneous environments, respectively. Building upon these approaches, Song et al developed a generalized transport equation with an environment-dependent diffusion kernel, providing a quantitative explanation for non-Fickian MSD and non-Gaussian displacement distribution. Based on the generalized transport equation, this study introduces an exactly solvable model for a non-Gaussian displacement distribution, accommodating arbitrary time profiles in its MSD, including both Fickian and non-Fickian behaviors. Our findings confirm the model’s capability in describing such transport processes. Furthermore, the proposed model unifies the CTRW model under fast environmental fluctuations and the SD model under Fickian time dependencies, making it suitable for understanding tracer particle motion within explicit solvent or complex media.

Upscaling transport in heterogeneous media featuring local-scale dispersion: Flow channeling, macro-retardation and parameter prediction

Many theoretical treatments of transport in heterogeneous Darcy flows consider advection only. When local-scale dispersion is neglected, flux weighting persists over time; mean Lagrangian and Eulerian flow velocity distributions relate simply to each other and to the variance of the underlying hydraulic conductivity field. Local-scale dispersion complicates this relationship, potentially causing initially flux-weighted solute to experience lower-velocity regions as well as Taylor-type macrodispersion due to transverse solute movement between adjacent streamlines. To investigate the interplay of local-scale dispersion with conductivity log-variance, correlation length, and anisotropy, we perform a Monte Carlo study of flow and advective-dispersive transport in spatially-periodic 2D Darcy flows in large-scale, high-resolution multivariate Gaussian random conductivity fields. We observe flow channeling at all heterogeneity levels and quantify its extent. We find evidence for substantial effective retardation in the upscaled system, associated with increased flow channeling, and observe limited Taylor-type macrodispersion, which we physically explain. A quasi-constant Lagrangian velocity is achieved within a short distance of release, allowing usage of a simplified continuous-time random walk (CTRW) model we previously proposed in which the transition time distribution is understood as a temporal mapping of unit time in an equivalent system with no flow heterogeneity. The numerical data set is modeled with such a CTRW; we show how dimensionless parameters defining the CTRW transition time distribution are predicted by dimensionless heterogeneity statistics and provide empirical equations for this purpose.

Predicting the Ki-67 proliferation index in cervical cancer: a preliminary comparative study of four non-Gaussian diffusion-weighted imaging models combined with histogram analysis.

The prognosis for patients with cervical cancer (CC) is strongly correlated with the Ki-67 proliferation index (PI). However, the Ki-67 PI obtained through biopsy has certain limitations. The non-Gaussian distribution diffusion model of magnetic resonance imaging (MRI) may play an important role in characterizing tissue heterogeneity. At present, there are limited data available concerning the prediction of Ki-67 PI using models based on histogram features of non-Gaussian diffusion distribution. This study aimed to determine whether preoperative histogram features from multiple non-Gaussian models of diffusion-weighted imaging can predict the Ki-67 PI in patients with CC. Our cross-sectional prospective study recruited a total of 53 patients suspected of having CC who underwent 3.0-T MRI at Sun Yat-sen Memorial Hospital of Sun Yat-sen University between January 2022 and January 2023. Fifteen b values (0-4,000 s/mm2) were used for diffusion-weighted imaging. A total of nine parameters from four non-Gaussian diffusion-weighted imaging models, including continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), fractional order calculus (FROC), and intravoxel incoherent motion (IVIM), were used. Whole-tumor volumetric histogram analysis of these parameters was then obtained. In logistic regression, significant histogram characteristics were statistically examined across two groups to build the final prediction model. To assess diagnostic parameters of the proposed model in the diagnosis of the Ki-67 PI, along with the sensitivity, specificity, and diagnostic accuracy of these various parameters from the four models, receiver operating feature analysis was applied. Among the 53 patients (55.3±9.6 years, ranging from 23 to 79 years) included in the study, 15 had a Ki-67 PI ≤50% and 38 had a Ki-67 PI >50%. Univariable analysis determined that 12 histogram features were statistically different between the two groups. In multivariable logistic regression, we ultimately selected 6 histogram features to construct the final prediction model, with CTRW_α_10th percentile [odds ratio (OR) =0.955; 95% confidence interval (CI): 0.92-0.99; P=0.019], CTRW_α_robust mean absolute deviation (OR =0.893; 95% CI: 0.81-0.99; P=0.028), and CTRW_α_uniformity (OR =0.000, 95% CI: 0.00-0.90, P=0.047) being the independent predictive variables. The area under the curve of the combined prediction model was 0.845 (95% CI: 0.74-0.95), with a sensitivity of 78.9% (95% CI: 0.63-0.90), a specificity of 86.7% (95% CI: 0.60-0.98), an accuracy of 81.1% (95% CI: 0.68-0.91), a positive predictive value of 93.8% (95% CI: 0.79-0.99), and a negative predictive value of 61.9% (95% CI: 0.38-0.82). The histogram features of multiple non-Gaussian diffusion-weighted imaging can help to predict the Ki-67 PI of CC, providing a new method for the noninvasive evaluation of critical biological features of CC.

Spatial patterns of the Brusselator model with asymmetric Lévy diffusion

Abstract The formation of spatial patterns is an important issue in reaction–diffusion systems. Previous studies have mainly focused on the spatial patterns in reaction–diffusion models equipped with symmetric diffusion (such as normal or fractional Laplace diffusion), namely, assuming that spatial environments of the systems are homogeneous. However, the complexity and heterogeneity of spatial environments of biochemical reactions in vivo can lead to asymmetric diffusion of reactants. Naturally, there arises an open question of how the asymmetric diffusion affects dynamical behaviors of biochemical reaction systems. To answer this, we build a general asymmetric Lévy diffusion model based on the theory of a continuous time random walk. In addition, we investigate the two-species Brusselator model with asymmetric Lévy diffusion, and obtain a general condition for the formation of Turing and wave patterns. More interestingly, we find that even though the Brusselator model with symmetric diffusion cannot produce steady spatial patterns for some parameters, the asymmetry of Lévy diffusion for this model can produce wave patterns. This is different from the previous result that wave instability requires at least a three-species model. In addition, the asymmetry of Lévy diffusion can significantly affect the amplitude and frequency of the spatial patterns. Our results enrich our knowledge of the mechanisms of pattern formation.

Effects of surface roughness and Reynolds number on the solute transport through three-dimensional rough-walled rock fractures under different flow regimes

In this study, the effects of surface roughness and Reynolds number (Re) on fluid flow and solute transport are investigated based on a double rough-walled fracture model that precisely represents the natural geometries of rock fractures. The double rough-walled fracture model is composed of two three-dimensional(3D) self-affine fracture surfaces generated using the improved successive random additions (SRA). Simulation of fluid flow and solute transport through the models were conducted by directly solving the Navier-Stokes equation and advection-diffusion equation (ADE), respectively. The results indicate that as the Re increases from 0.1 to 200, the flow regime changes from linear flow to nonlinear flow accompanied with the tortuous streamlines and significant eddies. Those eddies lead to the temporary stagnant zones that delay the solute migration. The increment of Re enhances the transport heterogeneity with the transport mode changing from the diffusion-dominated to the advection-dominated behavior, which is more significant in the fracture with a larger joint roughness coefficient (JRC). All breakthrough curves (BTCs) of rough-walled fractures exhibited typical non-Fickian transport characteristics with “early arrival” and “long tailing” of BTCs. Increasing the JRC and/or Re will enhances the non-Fickian transport characteristics. The ADE model is able to accurately fit the numerical BTCs and residence time distributions (RTDs) at a low Re, but fails to capture the non-Fickian transport characteristics at a large Re. In contrast, the continuous time random walk (CTRW) model provides a better fit to the numerical simulation results over the whole range of Re. Whereas, the fitting error gradually increases with increasing Re.

Transience of continuous-time conservative random walks

Abstract We consider two continuous-time generalizations of conservative random walks introduced in Englander and Volkov (2022), an orthogonal and a spherically symmetrical one; the latter model is also known as random flights. For both models, we show the transience of the walks when $d\ge 2$ and that the rate of direction changing follows a power law $t^{-\alpha}$ , $0<\alpha\le 1$ , or the law $(\!\ln t)^{-\beta}$ where $\beta>2$ .

Advanced diffusion-weighted MRI models for preoperative prediction of lymph node metastasis in resectable gastric cancer.

To investigate the potential of six advanced diffusion-weighted imaging (DWI) models for preoperative prediction of lymph node metastasis (LNM) in resectable gastric cancer (GC). Between Nov 2022 and Nov 2023, standard MRI scans were prospectively performed in consecutive patients with endoscopic pathology-confirmed gastric adenocarcinoma who were referred for direct radical gastrectomy. Six DWI models, including fractional order calculus (FROC), continuous-time random walk (CTRW), diffusion kurtosis imaging (DKI), intravoxel incoherent motion (IVIM), the mono-exponential model (MEM) and the stretched exponential model (SEM) were computed. Surgical pathologic diagnosis of LNM was the reference standard, and patients were classified into LNM-positive or LNM-negative groups accordingly. The morphological features and quantitative parameters of the DWI models in different LNM categories were analyzed and compared. Multivariable logistic regression was used to screen significant predictors. Receiver-operating characteristic curves and the area under the curve (AUC) were plotted to evaluate the performances, the Delong test was performed to compare the AUCs. In the LNM-positive group, tumor thickness and kurtosis (DKI_K) were significantly higher, while anomalous diffusion coefficient (CTRW_D), diffusivity (DKI_D), diffusion coefficient (FROC_D), pseudodiffusion coefficient (IVIM_D*), perfusion fraction (IVIM_f), and ADC were lower compared to the LNM-negative group. Clinical tumor staging (cT) and CTRW_D were independent predictors. Their combination demonstrated a superior AUC of 0.930, significantly higher than that of individual parameters. Tumor thickness, DKI_K, CTRW_D, DKI_D, FROC_D, IVIM_D*, IVIM_f and ADC were associated with LNM status. The combination of independent predictors of cT and CTRW_D further enhanced the performance.

Simulation of the continuous time random walk using subordination schemes.

The continuous time random walk model has been widely applied in various fields, including physics, biology, chemistry, finance, social phenomena, etc. In this work, we present an algorithm that utilizes a subordinate formula to generate data of the continuous time random walk in the long time limit. The algorithm has been validated using commonly employed observables, such as typical fluctuations of the positional distribution, rare fluctuations, the mean and the variance of the position, and breakthrough curves with time-dependent bias, demonstrating a perfect match.

Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes

Abstract We consider a generic one-dimensional stochastic process x(t), or a random walk Xn , which describes the position of a particle evolving inside an interval [ a , b ] , with absorbing walls located at a and b. In continuous time, x(t) is driven by some equilibrium process θ ( t ) , while in discrete time, the jumps of Xn follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability E b ( x , t ) , which is the probability for the particle to be absorbed first at the wall b, before or at time t, given its initial position x. In this paper we show that the derivation of this quantity can be tackled by studying a dual process y(t) very similar to x(t) but with hard walls at a and b. More precisely, we show that the quantity E b ( x , t ) for the process x(t) is equal to the probability Φ ~ ( x , t | b ) of finding the dual process inside the interval [ a , x ] at time t, with y ( 0 ) = b . This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.

From Lévy walks to fractional material derivative: Pointwise representation and a numerical scheme

The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier–Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions.We consider several typical differential equations involving the fractional material derivative and provide conditions for their solutions to exist. In some cases, the analytical solution can be found. For the general initial value problem, we devise a finite volume method and prove its stability, convergence, and conservation of probability. Numerical illustrations verify our analytical findings. Moreover, our numerical experiments show superiority in the computation time of the proposed numerical scheme over a Monte Carlo method applied to the problem of probability density function’s derivation.

First-passage and first-arrival problems in continuous-time random walks: Beyond the diffusion approximation.

Some exact solutions of the first-passage and first-arrival problems for the continuous-time random-walk model are obtained. On the basis of these exact solutions, the following has been revealed. First, for some jump-length distributions with a finite variance, the approximate solutions obtained in the diffusion approximation can differ significantly from the exact solutions. Second, for some waiting time distributions with a finite mean, the times of first passage and the times of first arrival can significantly depend on the ensemble under consideration. In particular, the mean first-passage time corresponding to the stationary ensemble can be significantly greater than the mean first-passage time corresponding to the nonaged ensemble. Third, for any continuous distribution of jump lengths, the probability of first arrival is zero for a point-like target. This last result is contrary to existing opinion, but it is consistent with the fact that a single point has a probability measure equal to zero in the probability space defined by a continuous distribution of jump lengths.

Multiple diffusion models for predicting pathologic response of esophageal squamous cell carcinoma to neoadjuvant chemotherapy.

To assess the feasibility and diagnostic performance of the fractional order calculus (FROC), continuous-time random-walk (CTRW), diffusion kurtosis imaging (DKI), intravoxel incoherent motion (IVIM), mono-exponential (MEM) and stretched exponential models (SEM) for predicting response to neoadjuvant chemotherapy (NACT) in patients with esophageal squamous cell carcinoma (ESCC). This study prospectively included consecutive ESCC patients with baseline and follow up MR imaging and pathologically confirmed cT1-4aN + M0 or T3-4aN0M0 and underwent radical resection after neoadjuvant chemotherapy (NACT) between July 2019 and January 2023. Patients were divided into pCR (TRG 0) and non-pCR (TRG1 + 2 + 3) groups according to tumor regression grading (TRG). The Pre-, Post- and Delta-treatment models were built. 18 predictive models were generated according to different feature categories, based on six models by five-fold cross-validation. Areas under the curve (AUCs) of the models were compared by using DeLong method. Overall, 90 patients (71 men, 19 women; mean age, 64years ± 6 [SD]) received NACT and underwent baseline and Post-NACT esophageal MRI, with 29 patients in the pCR group and 61 patients in the non-pCR group. Among18 predictive models, The Pre-, Post-, and Delta-CTRW model showed good predictive efficacy (AUC = 0.722, 0.833 and 0.790). Additionally, the Post-FROC model (AUC = 0.907) also exhibited good diagnostic performance. Our study indicates that the CTRW model, along with the Post-FROC model, holds significant promise for the future of NACT efficacy prediction in ESCC patients.

Whole-tumor histogram analysis of multiple non-Gaussian diffusion models at high b values for assessing cervical cancer.

To assess the diagnostic potential of whole-tumor histogram analysis of multiple non-Gaussian diffusion models for differentiating cervical cancer (CC) aggressive status regarding of pathological types, differentiation degree, stage, and p16 expression. Patients were enrolled in this prospective single-center study from March 2022 to July 2023. Diffusion-weighted images (DWI) were obtained including 15 b-values (0 ~ 4000s/mm2). Diffusion parameters derived from four non-Gaussian diffusion models including continuous-time random-walk (CTRW), diffusion-kurtosis imaging (DKI), fractional order calculus (FROC), and intravoxel incoherent motion (IVIM) were calculated, and their histogram features were analyzed. To select the most significant features and establish predictive models, univariate analysis and multivariate logistic regression were performed. Finally, we evaluated the diagnostic performance of our models by using receiver operating characteristic (ROC) analyses. 89 women (mean age, 55 ± 11 years) with CC were enrolled in our study. The combined model, which incorporated the CTRW, DKI, FROC, and IVIM diffusion models, offered a significantly higher AUC than that from any individual models (0.836 vs. 0.664, 0.642, 0.651, 0.649, respectively; p < 0.05) in distinguishing cervical squamous cell cancer from cervical adenocarcinoma. To distinguish tumor differentiation degree, except the combined model showed a better predictive performance compared to the DKI model (AUC, 0.839 vs. 0.697, respectively; p < 0.05), no significant differences in AUCs were found among other individual models and combined model. To predict the International Federation of Gynecology and Obstetrics (FIGO) stage, only DKI and FROC model were established and there was no significant difference in predictive performance among different models. In terms of predicting p16 expression, the predictive ability of DKI model is significantly lower than that of FROC and combined model (AUC, 0.693 vs. 0.850, 0.859, respectively; p < 0.05). Multiple non-Gaussian diffusion models with whole-tumor histogram analysis show great promise to assess the aggressive status of CC.

Aging and confinement in subordinated fractional Brownian motion.

We study the effects of aging properties of subordinated fractional Brownian motion (FBM) with drift and in harmonic confinement, when the measurement of the stochastic process starts a time t_{a}>0 after its original initiation at t=0. Specifically, we consider the aged versions of the ensemble mean-squared displacement (MSD) and the time-averaged MSD (TAMSD), along with the aging factor. Our results are favorably compared with simulations results. The aging subordinated FBM exhibits a disparity between MSD and TAMSD and is thus weakly nonergodic, while strong aging is shown to effect a convergence of the MSD and TAMSD. The information on the aging factor with respect to the lag time exhibits an identical form to the aging behavior of subdiffusive continuous-time random walks (CTRW). The statistical properties of the MSD and TAMSD for the confined subordinated FBM are also derived. At long times, the MSD in the harmonic potential has a stationary value, that depends on the Hurst index of the parental (nonequilibrium) FBM. The TAMSD of confined subordinated FBM does not relax to a stationary value but increases sublinearly with lag time, analogously to confined CTRW. Specifically, short aging times t_{a} in confined subordinated FBM do not affect the aged MSD, while for long aging times the aged MSD has a power-law increase and is identical to the aged TAMSD.

Non-Gaussian diffusion metrics with whole-tumor histogram analysis for bladder cancer diagnosis: muscle invasion and histological grade

PurposeTo investigate the performance of histogram features of non-Gaussian diffusion metrics for diagnosing muscle invasion and histological grade in bladder cancer (BCa).MethodsPatients were prospectively allocated to MR scanner1 (training cohort) or MR2 (testing cohort) for conventional diffusion-weighted imaging (DWIconv) and multi-b-value DWI. Metrics of continuous time random walk (CTRW), diffusion kurtosis imaging (DKI), fractional-order calculus (FROC), intravoxel incoherent motion (IVIM), and stretched exponential model (SEM) were simultaneously calculated using multi-b-value DWI. Whole-tumor histogram features were extracted from DWIconv and non-Gaussian diffusion metrics for logistic regression analysis to develop diffusion models diagnosing muscle invasion and histological grade. The models’ performances were quantified by area under the receiver operating characteristic curve (AUC).ResultsMR1 included 267 pathologically-confirmed BCa patients (median age, 67 years [IQR, 46–82], 222 men) and MR2 included 83 (median age, 65 years [IQR, 31–82], 73 men). For discriminating muscle invasion, CTRW achieved the highest testing AUC of 0.915, higher than DWIconv’s 0.805 (p = 0.014), and similar to the combined diffusion model’s AUC of 0.885 (p = 0.076). For differentiating histological grade of non-muscle-invasion bladder cancer, IVIM outperformed a testing AUC of 0.897, higher than DWIconv’s 0.694 (p = 0.020), and similar to the combined diffusion model’s AUC of 0.917 (p = 0.650). In both tasks, DKI, FROC, and SEM failed to show diagnostic superiority over DWIconv (p > 0.05).ConclusionCTRW and IVIM are two potential non-Gaussian diffusion models to improve the MRI application in assessing muscle invasion and histological grade of BCa, respectively.Critical relevance statementOur study validates non-Gaussian diffusion imaging as a reliable, non-invasive technique for early assessment of muscle invasion and histological grade in BCa, enhancing accuracy in diagnosis and improving MRI application in BCa diagnostic procedures.Key PointsMuscular invasion largely determines bladder salvageability in bladder cancer patients.Evaluated non-Gaussian diffusion metrics surpassed DWIconv in BCa muscle invasion and histological grade diagnosis.Non-Gaussian diffusion imaging improved MRI application in preoperative diagnosis of BCa.Graphical

Laplace’s first law of errors applied to diffusive motion

In biological, glassy, and active systems, various tracers exhibit Laplace-like, i.e., exponential, spreading of the diffusing packet of particles. The limitations of the central limit theorem in fully capturing the behaviors of such diffusive processes, especially in the tails, have been studied using the continuous time random walk model. For cases when the jump length distribution is super-exponential, e.g., a Gaussian, we use large deviations theory and relate it to the appearance of exponential tails. When the jump length distribution is sub-exponential, the packet of spreading particles is described by the big jump principle. We demonstrate the applicability of our approach for finite time, indicating that rare events and the asymptotics of the large deviations rate function can be sampled for large length scales within a reasonably short measurement time.Graphical abstractThe universality of Laplace tails appears everywhere

Utilization of diffusion-weighted derived mathematical models to predict prognostic factors of resectable rectal cancer.

This study explored models of monoexponential diffusion-weighted imaging (DWI), diffusion kurtosis imaging (DKI), stretched exponential (SEM), fractional-order calculus (FROC), and continuous-time random-walk (CTRW) as diagnostic tools for assessing pathological prognostic factors in patients with resectable rectal cancer (RRC). RRC patients who underwent radical surgery were included. The apparent diffusion coefficient (ADC), the mean kurtosis (MK) and mean diffusion (MD) from the DKI model, the distributed diffusion coefficient (DDC) and α from the SEM model, D, β and u from the FROC model, and D, α and β from the CTRW model were assessed. There were a total of 181 patients. The area under the receiver operating characteristic (ROC) curve (AUC) of CTRW-α for predicting histology type was significantly higher than that of FROC-u (0.780 vs. 0.671, p = 0.043). The AUC of CTRW-α for predicting pT stage was significantly higher than that of FROC-u and ADC (0.786 vs.0.683, p = 0.043; 0.786 vs. 0.682, p = 0.030), the difference in predictive efficacy of FROC-u between ADC and MK was not statistically significant [0.683 vs. 0.682, p = 0.981; 0.683 vs. 0.703, p = 0.720]; the difference between the predictive efficacy of MK and ADC was not statistically significant (p = 0.696). The AUC of CTRW (α + β) (0.781) was significantly higher than that of FROC-u (0.781 vs. 0.625, p = 0.003) in predicting pN stage but not significantly different from that of MK (p = 0.108). The CTRW and DKI models may serve as imaging biomarkers to predict pathological prognostic factors in RRC patients before surgery.

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.

Time-dependent dispersion coefficients for the evolution of displacement fronts in heterogeneous porous media

We present an approach for quantifying displacement fronts in heterogeneous porous media based on the concept of time-dependent apparent dispersion coefficients. The concept of constant asymptotic macrodispersion generally overestimates the area swept by a displacement front and leads to unrealistic upstream dispersion. We show that the large-scale front spreading can be captured by a one-dimensional advection–dispersion equation that is parameterized by a suitably chosen temporally evolving dispersion coefficient. For purely advective front spreading, we derive an analytical expression based on a predictive continuous time random walk approach, which applies to highly heterogeneous porous media. This analysis elucidates the variability of solute travel times as the key longitudinal spreading mechanism. It shows that the evolution of dispersion can be captured as the sum of exponentials that decay on two dominant time scales. In a particle-based picture, these scales mark the short time at which transported particles start exploring the flow variability and the large time at which the slowest particles start decorrelating their transport velocity. Based on these insights, we propose a heuristic formula that accounts for the impact of local-scale dispersion as an additional decorrelation mechanism. The heuristic expression for the longitudinal dispersion coefficient captures solute spreading for a broad range of Péclet numbers and heterogeneity variances. The proposed approach is tested against direct numerical simulations. It provides a robust and fast method for quantifying the evolution of displacement fronts in heterogeneous porous media with possible applications, for example, in groundwater contamination modelling, underground gas storage, and geothermal energy production.

Ultrasound enhanced diffusion in hydrogels: An experimental and non-equilibrium molecular dynamics study.

Focused ultrasound has experimentally been found to enhance the diffusion of nanoparticles; our aim with this work is to study this effect closer using both experiments and non-equilibrium molecular dynamics. Measurements from single particle tracking of 40nm polystyrene nanoparticles in an agarose hydrogel with and without focused ultrasound are presented and compared with a previous experimental study using 100nm polystyrene nanoparticles. In both cases, we observed an increase in the mean square displacement during focused ultrasound treatment. We developed a coarse-grained non-equilibrium molecular dynamics model with an implicit solvent to investigate the increase in the mean square displacement and its frequency and amplitude dependencies. This model consists of polymer fibers and two sizes of nanoparticles, and the effect of the focused ultrasound was modeled as an external oscillating force field. A comparison between the simulation and experimental results shows similar mean square displacement trends, suggesting that the particle velocity is a significant contributor to the observed ultrasound-enhanced mean square displacement. The resulting diffusion coefficients from the model are compared to the diffusion equation for a two-time continuous time random walk. The model is found to have the same frequency dependency. At lower particle velocity amplitude values, the model has a quadratic relation with the particle velocity amplitude as described by the two-time continuous time random walk derived diffusion equation, but at higher amplitudes, the model deviates, and its diffusion coefficient reaches the non-hindered diffusion coefficient. This observation suggests that at higher ultrasound intensities in hydrogels, the non-hindered diffusion coefficient can be used.