Time-dependent Diffusion Coefficient Research Articles

Microrheological model for Kelvin-Voigt materials with micro-heterogeneities.

We introduce a generalization of the Kelvin-Voigt model in order to include and characterize heterogeneities in viscoelastic semisolid materials. By considering a microrheological approach, we present analytical expressions for the mean square displacement and for the time-dependent diffusion coefficient of probe particles immersed in a viscoelastic material described by this model. Besides validating our theoretical approach through Brownian dynamics simulations, we show how the model can be used to describe experimental data obtained for polyacrylamide and LAPONITE® gels.

Long-range hot carrier transport in lead-free cesium tin halide perovskite microplates

The carrier dynamics of lead-free cesium tin halide perovskites are crucial for evaluating their potential as substitutes for lead-based perovskites. Herein, we investigate the ultrafast transport of hot carriers in CsSnBr3 microplates synthesized via chemical vapor deposition. We observe a significantly prolonged hot carrier cooling process lasting ∼50 ps, due to the hot-phonon bottleneck effect and the formation of large polarons. By employing transient absorption microscopy, we detect a time-dependent diffusion coefficient of hot carriers initially reaching up to 257.8 cm2·s−1, leading to a long average travel distance of ∼600 nm. Subsequently, it reveals a remarkable equilibrium carrier diffusion coefficient ranging from 12.7 to 16.5 cm2·s−1. These findings establish a fundamental basis for designing and advancing lead-free perovskite-based optoelectronic applications involving hot carriers.

Fast and Accurate Numerical Integration of the Langevin Equation with Multiplicative Gaussian White Noise.

A univariate stochastic system driven by multiplicative Gaussian white noise is considered. The standard method for simulating its Langevin equation of motion involves incrementing the system's state variable by a biased Gaussian random number at each time step. It is shown that the efficiency of such simulations can be significantly enhanced by incorporating the skewness of the distribution of the updated state variable. A new algorithm based on this principle is introduced, and its superior performance is demonstrated using a model of free diffusion of a Brownian particle with a friction coefficient that decreases exponentially with the kinetic energy. The proposed simulation technique proves to be accurate over time steps that are an order of magnitude longer than those required by standard algorithms. The model used to test the new numerical technique is known to exhibit a transition from normal diffusion to superdiffusion as the environmental temperature rises above a certain critical value. A simple empirical formula for the time-dependent diffusion coefficient, which covers both diffusion regimes, is introduced, and its accuracy is confirmed through comparison with the simulation results.

Charge transport in organic semiconductors from the mapping approach to surface hopping.

We describe how to simulate charge diffusion in organic semiconductors using a recently introduced mixed quantum-classical method, the mapping approach to surface hopping. In contrast to standard fewest-switches surface hopping, this method propagates the classical degrees of freedom deterministically on the most populated adiabatic electronic state. This correctly preserves the equilibrium distribution of a quantum charge coupled to classical phonons, allowing one to time-average along trajectories to improve the statistical convergence of the calculation. We illustrate the method with an application to a standard model for the charge transport in the direction of maximum mobility in crystalline rubrene. Because of its consistency with the equilibrium distribution, the present method gives a time-dependent diffusion coefficient that plateaus correctly to a long-time limiting value. The resulting mobility is somewhat higher than that of the relaxation time approximation, which uses a phenomenological relaxation parameter to obtain a non-zero diffusion coefficient from a calculation with static phonon disorder. However, it is very similar to the mobility obtained from Ehrenfest dynamics, at least in the parameter regimes we have investigated here. This is somewhat surprising because Ehrenfest dynamics overheats the electronic subsystem and is, therefore, inconsistent with the equilibrium distribution.

Time-dependent gas dynamic diffusion process in shale matrix: model development and numerical analysis

Gas diffusion is a pivotal process during shale gas recovery, which is determined by diffusion coefficient to a large extent. In previous studies, the gas diffusion coefficient is generally assumed as a constant. However, increasing experiments prove that the diffusion coefficient of shale gas is strongly time-dependent. Therefore, to perfect the theory of shale gas diffusion, this paper proposes a time-dependent diffusion model for shale gas, which incorporates time-dependent gas diffusion coefficient, composing of the bulk diffusion coefficient for free gas in organic and inorganic pores, as well as the surface diffusion coefficient for adsorbed gas in organic pores. To validate the accuracy of the new theory, we calibrate the theoretical results against experimental data, and the results show that they have strong correlation, and the time-dependent diffusion model is superior to classical model. Finally, the numerical analysis of gas dynamic diffusion process in shale matrix is conducted. The results show that at the end of diffusion, a large amounts of shale gas remain trapped in the matrix core due to the attenuation of gas diffusion coefficient. In addition, neglecting the time-dependent nature of gas diffusion in shale matrix leads to a significant overestimation of gas production.

Similarity analysis of chloride transport behavior in fly ash concrete under different environments aiding by machine learning method

To forecast chloride transport in site by applying accelerated test results, it is necessary to investigate the similarity of chloride diffusivity under different environments. However, limited by test period and cost expenses, a large amount of tests in field and simulated environment are hard to conduct, and a limit test data will make prediction on chloride diffusivity of concrete less precise. To solve this problem, machine learning methods are explored to predict chloride transport. Based on the existing chloride concentration test data of fly ash concrete in natural and accelerated simulated environment, this paper establishes a chloride concentration predictive model based on the one-dimensional convolutional neural network (1D-CNN) method at first, and verifies the stability and accuracy of the established model. Second, similarity models for concentration ratio and diffusion time ratio based on multivariate nonlinear model and based on theoretical Fick’s second law are constructed, and the model accuracy is also tested. Finally, the chloride concentration data predicted by the 1D-CNN model is applied to the constructed similarity models respectively, and the application results are compared and analyzed to determine the optimal similarity model. Results indicate that the concentration predicted by the 1D-CNN method has a strong robustness and a small prediction error. Besides, based on multivariate nonlinear model, the error of concentration predicted by the similarity models for concentration ratio and diffusion time ratio is less than 16 % and 30 %, respectively. Similarly, according to the theoretical model based on Fick’s second law, the error of concentration predicted by the similarity models for concentration ratio based on the time dependent chloride diffusion coefficient and time dependent peak chloride concentration is less than 21 % and 36 %, respectively. The similarity model for concentration ratio based on multivariate nonlinear model has the best prediction effect.

Fully discrete Galerkin scheme for a semilinear subdiffusion equation with nonsmooth data and time-dependent coefficient

We couple the L1 discretization of the Caputo fractional derivative in time with the Galerkin scheme to devise a linear numerical method for the semilinear subdiffusion equation. Two important points that we make are nonsmooth initial data and time-dependent diffusion coefficient. We prove the stability and convergence of the method under weak assumptions concerning regularity of the diffusivity. We find optimal pointwise in space and global in time errors, which are verified with several numerical experiments.

Diffusing Wave Microrheology in Polymeric Fluids.

Recently, there has been interest in determining the viscoelastic properties of polymeric liquids and other complex fluids by means of Diffusing Wave Spectroscopy (DWS). In this technique, light-scattering spectroscopy is applied to highly turbid fluids containing optical probe particles. The DWS spectrum is used to infer the time-dependent mean-square displacement and time-dependent diffusion coefficient D of the probes. From D, values for the storage modulus G'(ω) and the loss modulus G''(ω) are obtained. This paper is primarily concerned with the inference of the mean-square displacement from a DWS spectrum. However, in much of the literature, central to the inference that is said to yield D is an invocation g(1)(t)=exp(-2q2X(t)2¯) of the Gaussian Approximation for the field correlation function g(1)(t) of the scattered light in terms of the mean-square displacement X(t)2¯ of a probe particle during time t. Experiment and simulation both show that the Gaussian approximation is invalid for probes in polymeric liquids and other complex fluids. In this paper, we obtain corrections to the Gaussian approximation that will assist in interpreting DWS spectra of probes in polymeric liquids. The corrections reveal that these DWS spectra receive contributions from higher moments X(t)2n¯, n>1, of the probe displacement distribution function.

Time-dependent reliability assessment and optimal design of corroded reinforced concrete beams

It remains a significant challenge to quantitatively describe the corrosion of reinforced concrete (RC) structures under chloride penetration. Moreover, when considering uncertainties throughout the life cycle of corroded RC structures for assessing their safety, reliability, and optimal design, the complexity of the problem intensifies. To address these issues, this paper studies the time-dependent reliability analysis and optimal design of corroded RC beams. At first, the time-dependent reliability of beams is investigated by considering both the serviceability limit state (SLS), which corresponds to the corrosion initiation of the reinforced steel, and the ultimate limit state (ULS), associated with the bending failure of the beam. This analysis takes into account the time-dependent chloride diffusion coefficient and incorporates a stochastic process. The reliability is evaluated using the Monte Carlo Simulation (MCS) method and the cumulative distribution function (CDF) method. Subsequently, a time-dependent reliability-based design optimization (TRBDO) problem is formulated, and the PSO-MCS, a methodology incorporating a particle swarm optimization (PSO) algorithm and MCS is adopted to solve it. After optimization, the initial cost of the specific RC beam is reduced from 1351.879€ to 1247.075€, while the time-dependent reliability within [0, 100] years is improved from 0.6057 to 0.6508. The effectiveness of the CDF, MCS and PSO-MCS methods are demonstrated through reliability analysis and design examples of corroded RC beams.

Anomalous heat transport and universality in macroscopic diffusion models

Anomalous diffusion is ubiquitous in nature and relevant for a wide range of applications, including energy transport, especially in bio- and nano-technologies. Numerous approaches have been developed to describe it from a microscopic point of view, and recently, it has been framed within universality classes, characterized by the behaviour of the moments and auto-correlation functions of the transported quantities. It is important to investigate whether such universality applies to macroscopic models. Here, the spectrum of the moments of the solutions of the transport equations is investigated for three continuous PDE models featuring anomalous diffusion. In particular, we consider the transport described by: (i) a generalized diffusion equation with time-dependent diffusion coefficient; (ii) the Porous Medium Equation and (iii) the Telegrapher Equation. For each model, the key features of the source-type solution as well as the analytical results for the moment analysis are revisited and extended via both analytical and numerical approaches. Equivalence of the asymptotic behaviour of the corresponding heat transport is confirmed within the realm of weak anomalous diffusion.

Thermodynamic description of active brownian particle driven by fractional gaussian noise

As a natural extension of the recent results on the thermodynamics of an active Brownian particle (self-propelled), we study the thermodynamics of an active Brownian particle (ABP) driven by fractional Gaussian noise (FGN). To serve as a prelude of the main results, we start from the conventional Markov process but with time dependent diffusion coefficient, where deviation in integral fluctuation relation (IFR) for total entropy production requires a general definition of the temperature, following the same case for a Brownian particle. In other words, the general temperature definition for this case is independent to the statistics of the rotational motion. We then proceed with the main problem of the paper, which is an active Brownian particle driven by fractional Gaussian noise. Under the assumption that self-propulsion is even under time-reversal, temperature is defined as well as the distance on how far the IFR for total entropy production deviates from the standard definition by adopting the standard definition of trajectory-level entropy and the joint probability of ABP. Furthermore, second law-like concept based on the found deviation is derived, as well as a generalized Clausius inequality. Lastly, magnitude of this deviation diminishes in the case of pure white noise.

Exact higher-order moments for linear non-homogeneous stochastic differential equation

This paper investigates the moments of a stochastic process that satisfies the one-dimensional linear stochastic differential equation (SDE) with nonlinear time-dependent drift and diffusion coefficients. The goal is to derive formulas for the nth exact moment, that instead of seeking the transition density function by solving the Fokker-Plank equations or moment-generating functions, which can be difficult to solve in closed form. We will appropriately apply Itô’s formula and the properties of the Wiener process with a constant drift and diffusion term, which is a Gaussian process to obtain the exact higher-order moments.

Tempered fractional diffusion in comb-like structures with numerical investigation

This paper presents two models for describing anomalous transport in comb-like structures. First, we analytically solve the tempered fractional diffusion model using the Laplace-Fourier technique. The probability distributions along the backbone (x-axis) and branches (y-axis) are represented by the M-Wright and Fox’s H functions. The probability distributions are illustrated according to the order of the time-fractional derivative α and the so-called tempered parameter λ. Additionally, we determine the mean square displacement to classify the degree of diffusivity in the comb structure based on the values of the time-fractional and tempered orders. Second, we introduce a power-law time-dependent diffusion coefficient as an extension of the comb-like models and investigate the solution of via numerical simulation. Then, we explore the connection between the presence of a time-dependent diffusion coefficient and anomalous transport based on the particle density and mean square displacement.

Fractional and scaled Brownian motion on the sphere: The effects of long-time correlations on navigation strategies.

We analyze fractional Brownian motion and scaled Brownian motion on the two-dimensional sphere S^{2}. We find that the intrinsic long-time correlations that characterize fractional Brownian motion collude with the specific dynamics (navigation strategies) carried out on the surface giving rise to rich transport properties. We focus our study on two classes of navigation strategies: one induced by a specific set of coordinates chosen for S^{2} (we have chosen the spherical ones in the present analysis), for which we find that contrary to what occurs in the absence of such long-time correlations, nonequilibrium stationary distributions are attained. These results resemble those reported in confined flat spaces in one and two dimensions [Guggenberger et al. New J. Phys. 21, 022002 (2019)1367-263010.1088/1367-2630/ab075f; Vojta et al. Phys. Rev. E 102, 032108 (2020)2470-004510.1103/PhysRevE.102.032108]; however, in the case analyzed here, there are no boundaries that affect the motion on the sphere. In contrast, when the navigation strategy chosen corresponds to a frame of reference moving with the particle (a Frenet-Serret reference system), then the equilibrium distribution on the sphere is recovered in the long-time limit. For both navigation strategies, the relaxation times toward the stationary distribution depend on the particular value of the Hurst parameter. We also show that on S^{2}, scaled Brownian motion, distinguished by a time-dependent diffusion coefficient with a power-scaling, is independent of the navigation strategy finding a good agreement between the analytical calculations obtained from the solution of a time-dependent diffusion equationon S^{2}, and the numerical results obtained from our numerical method to generate ensemble of trajectories.

Rayleigh–Bénard instability in a horizontal porous layer with anomalous diffusion

The analysis of the Rayleigh–Bénard instability due to the mass diffusion in a fluid-saturated horizontal porous layer is reconsidered. The standard diffusion theory based on the variance of the molecular position growing linearly in time is generalized to anomalous diffusion, where the variance is modeled as a power-law function of time. A model of anomalous diffusion based on a time-dependent mass diffusion coefficient is adopted, together with Darcy's law, for momentum transfer, and the Boussinesq approximation, for the description of the buoyant flow. A linear stability analysis is carried out for a basic state where the solute has a potentially unstable concentration distribution varying linearly in the vertical direction, and the fluid is at rest. It is shown that any, even slight, departure from the standard diffusion process has a dramatic effect on the onset conditions of the instability. This circumstance reveals a strong sensitivity to the anomalous diffusion index. It is shown that subdiffusion yields instability for every positive mass diffusion Rayleigh number, while superdiffusion brings stabilization no matter how large is the Rayleigh number. A discussion of the linear stability analysis based on the Galilei-variant fractional-derivative model of subdiffusion is eventually carried out.

Direct method to estimate the gas loss characteristics and in-situ gas contents of shale

The commonly used direct methods for the lost gas and in-situ gas content estimation of shale usually follow the assumptions and mathematical models that were originally established for coalbed. The essential differences in formation fluid compositions, coring and on-site desorption processes, governing equations on gas release, and model parameters between shale and coal are not comprehensively analyzed for these available direct methods, therefore, they cannot fully reflect the gas loss characteristics of shale and are also inaccurate for geological reserves calculation. In this paper, the modified lost gas content calculation method is constructed by analyzing the general coring and on-site desorption processes of shale. This method simultaneously depicts the loss characteristics of free and adsorbed gas. It can be summarized that the initial adsorbed gas concentration, time-dependent gas diffusion coefficient, time-dependent desorbed gas concentration at the core boundary, core size, porosity, methane saturation, formation temperature, and formation pressure are the influencing parameters for accurate in-situ shale gas content calculation. Herein, the estimation or optimization methods for some of these parameters are presented. The fitting results manifest that the newly presented method shows satisfactory applicability in gas loss curve prediction and in-situ gas content estimation based on single and twice heating on-site desorption data. It is found that some existing direct methods may exhibit anomalies or stability problems in shale gas loss curve characterization, therefore, the assumption reasonability and mathematical-physical law consistency of these methods should be further confirmed. By contrast, the assumptions of the new method are closer to the practical shale coring, especially the simultaneous loss of free and adsorbed gas as well as the introduction of time-dependent parameters, laying the foundation for a more accurate gas loss curve prediction and geological reserves calculation for shale.

Time-dependent reliability analysis of corroded RC beams based on the phase-type fitting method

It remains an important challenge to quantitatively describe the corrosion of reinforced concrete (RC) structures under chloride penetration. When considering the uncertainties encountered throughout the life cycle of RC structures exposed to a corrosive environment and evaluating their safety and reliability, the complexity of the problem intensifies. To address these issues, this paper focuses on the time-dependent reliability analysis of corroded RC beams, utilizing the phase-type (PH) fitting method. Initially, a model for the time-dependent reliability of corroded RC beams is established, incorporating the time-dependent chloride diffusion coefficient. Subsequently, a novel PH fitting method is proposed. The effectiveness of this new method is demonstrated through numerical examples. Furthermore, the time-dependent reliability analysis of corroded RC beams is compared using both the PH fitting method and the Monte Carlo simulation. The results reveal that the proposed method can accurately and efficiently deal with time-dependent reliability problems.

On the Inverse Problem of Time-Dependent Coefficient in a Time Fractional Diffusion Problem by Newly Defined Monic Laquerre Wavelets

Abstract The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.

Fractal derivative model with time dependent diffusion coefficient for chloride diffusion in concrete

More research and experiments have indicated that, under different water cement ratios, the binding effect of concrete on chloride ions and the blocking effect of hydration on chloride ions are clearly dominant. During this time period, the diffusion rate of chloride ions will increase or decrease with time, resulting in the classical Fickian diffusion law no longer being applicable. This study establishes an improved fractal model with a time-dependent diffusion coefficient for chloride diffusion in concrete. An explicit solution is derived in terms of the complementary error function. The proposed model is verified by fitting the chloride ion concentration data from different concrete immersion tests. The results show that the fractal model can fit the data in different spatial scales, and the corresponding prediction accuracy is rather high when compared to the three integer order models with constant, variable diffusion coefficient, and curing age. Therefore, the fractal derivative model is feasible in describing the anomalous diffusion of chloride ions in concrete.

Trends to equilibrium for a nonlocal Fokker–Planck equation

We obtain equilibration rates for a one-dimensional nonlocal Fokker–Planck equation with time-dependent diffusion coefficient and drift, modeling the relaxation of a large swarm of robots, feeling each other in terms of their distance, towards the steady profile characterized by uniform spreading over a finite interval of the line. The result follows by combining entropy methods for quantifying the decay of the solution towards its quasi-stationary distribution, with the properties of the quasi-stationary profile.