Power-law Diffusion Research Articles

A posteriori error analysis of a positivity preserving scheme for the power-law diffusion Keller–Segel model

Abstract We study a finite volume scheme approximating a parabolic-elliptic Keller–Segel system with power-law diffusion with exponent $\gamma \in [1,3]$ and periodic boundary conditions. We derive conditional a posteriori bounds for the error measured in the $L^{\infty }(0,T;H^{1}(\varOmega ))$ norm for the chemoattractant and by a quasi-norm-like quantity for the density. These results are based on stability estimates and suitable conforming reconstructions of the numerical solution. We perform numerical experiments showing that our error bounds are linear in mesh width and elucidating the behavior of the error estimator under changes of $\gamma $.

Discussion of “Water distribution characteristics of capillary absorption in internally cured concrete with superabsorbent polymers”

In March 2024 Construction and Building Materials published “Water distribution characteristics of capillary absorption in internally cured concrete with superabsorbent polymers”, which claims to present a water distribution model characterising the spatiotemporal moisture content evolution in internally cured concrete with superabsorbent polymers during capillary absorption. This discussion, considered a post-publication critique, establishes that the paper’s water distribution model conflicts with the paper’s capillary absorption tests. While the model is based on the premise of the square-root-of-time behaviour of capillary absorption, the tests do not demonstrate that square-root-of-time behaviour. This critique in addition questions the paper’s forthright application of the power-law diffusivity expression with exponent 4, given that its validity for the paper’s materials has not been verified. It is shown that different but equivalent diffusivity expressions lead to different moisture diffusivities at lower moisture contents.

Extraction of Bioactive Secondary Metabolites from Citrus Maxima Peel Via Pressurized Hot Water Extractor for the Synthesis of Iron Oxide Nanoparticles

Using bioactive secondary metabolites (BSM) from Citrus maxima peel extract, this work investigated a green method of synthesizing magnetite iron oxide (Fe3O4) nanoparticles (NPs). The BSM acts as a reducing cum stabilizing agent in the process. The optimization of the BSM parameters were conducted through response surface methodology (RSM). The optimal condition obtained in this study for pressurized hot water extractor (PHWE) extraction was temperature of 94.96 °C, solvent-to-solid ratio of 29.7 ml/g and extraction time of 27.6 min. BSM Yield of 49.31 % could be obtained based on this condition. The formation of Fe3O4 NPs were detected using the FTIR analysis with the absorption peaks observed at around 590 and 580 cm−1. X-ray diffraction results matched standard magnetite Fe3O4 patterns with planes at (220), (311), (400), (511) and (440). Field emission scanning electron microscopy (FESEM) results showed that the magnetite Fe3O4 NPs synthesized in this study predominantly appeared in spherical shape. The extraction process's kinetics were examined using various empirical models, including the Elovich’s model, Peleg’s model, Power law and parabolic diffusion model. All applied models were found to be well fitted with the measured data from experiment, with R2 values exceeding 0.9. Notably, the Peleg’s model exhibited the highest R2, the smallest RMSD, and the least significant p-values, indicating its superior performance.

Ultrafast Diffusion Modeling via the Riemann–Liouville Nonlocal Structural Derivative and Its Application in Porous Media

Ultrafast diffusion disperses faster than super-diffusion, and this has been proven by several theoretical and experimental investigations. The mean square displacement of ultrafast diffusion grows exponentially, which provides a significant challenge for modeling. Due to the inhomogeneity, nonlinear interactions, and high porosity of cement materials, the motion of particles on their surfaces satisfies the conditions for ultrafast diffusion. The investigation of the diffusion behavior in cementitious materials is crucial for predicting the mechanical properties of cement. In this study, we first attempted to investigate the dynamic of ultrafast diffusion in cementitious materials underlying the Riemann–Liouville nonlocal structural derivative. We constructed a Riemann–Liouville nonlocal structural derivative ultrafast diffusion model with an exponential function and then extended the modeling strategy using the Mittag–Leffler function. The mean square displacement is analogous to the integral of the corresponding structural derivative, providing a reference standard for the selection of structural functions in practical applications. Based on experimental data on cement mortar, the accuracy of the Riemann–Liouville nonlocal structural derivative ultrafast diffusion model was verified. Compared to the power law diffusion and the exponential law diffusion, the mean square displacement with respect to the Mittag–Leffler law is closely tied to the actual data. The modeling approach based on the Riemann–Liouville nonlocal structural derivative provides an efficient tool for depicting ultrafast diffusion in porous media.

Weibull dynamics and power-law diffusion of epidemics in small world 2D networks

We demonstrate that the Weibull statistics characterize the time dependence of the epidemic spread in several idealized cases on 2-D Cartesian lattices, with and without small world structure. These cases include infection from nearest and next nearest neighbors; infection from long-distant neighbors; infection due to external influence. The Weibull closed form for the aggregated disease propagation is used as an analytic tool to elucidate the effect of small world topology in the diffusion scaling. It may be also used to formalize and contain the extensive diversity of real life cases. Along these lines, besides the straightforward implementation of the studied cases in epidemiology, our work proposes a power-law scaling in the Bass model for applications in the diffusion of the technological innovation.

Mathematical Modeling Thoughts and Methods Based on Fractional Differential Equations in Teaching

Abstract This article combines ordinary differential equations’ theoretical and practical characteristics to explore how to integrate mathematical modeling ideas in teaching materials and teaching methods. We apply the fractional differential equation algorithm to the mathematical modeling of Terman diffusion. At the same time, the article proposes a framework for obtaining accurate solutions of a class of nonlinear Sinai stochastic models with power-law diffusion through Mittag-Leffler function transformation. In this way, we have obtained great explicit and accurate solutions of some important Mittag-Leffler function power-law diffusion physical processes.

Efficiency of random search with space-dependent diffusivity.

We address the problem of random search for a target in an environment with a space-dependent diffusion coefficient D(x). Considering a general form of the diffusion differential operator that includes Itô, Stratonovich, and Hänggi-Klimontovich interpretations of the associated stochastic process, we obtain and analyze the first-passage-time distribution and use it to compute the search efficiency E=〈1/t〉. For the paradigmatic power-law diffusion coefficient D(x)=D_{0}|x|^{α}, where x is the distance from the target and α<2, we show the impact of the different interpretations. For the Stratonovich framework, we obtain a closed-form expression for E, valid for arbitrary diffusion coefficient D(x). This result depends only on the distribution of diffusivity values and not on its spatial organization. Furthermore, the analytical expression predicts that a heterogeneous diffusivity profile leads to a lower efficiency than the homogeneous one with the same average level within the space between the target and the searcher initial position, but this efficiency can be exceeded for other interpretations.

Monte Carlo Study of Electron and Positron Cosmic-Ray Propagation with the CALET Spectrum

Abstract Focusing on the electron and positron spectrum measured with the Calorimetric Electron Telescope (CALET), which shows characteristic structures, we calculate the flux contributions of cosmic rays that have escaped from randomly appearing supernova remnants. We adopt a Monte Carlo method to take into account the stochastic nature of the appearance of nearby sources. We find that without a complicated energy dependence of the diffusion coefficient, simple power-law diffusion coefficients can produce spectra similar to the CALET spectrum, even with a dispersion in the injection index. The positron component measured with AMS-02 is consistent with a bump-like structure around 300 GeV in the CALET spectrum. One to three nearby supernovae can contribute up to a few tens of percent of the CALET flux at 2–4 TeV, while ten or more unknown and distant (≳500 pc) supernovae account for the remaining several tens of percent of the flux. The CALET spectrum, showing a sharp drop at ∼1 TeV, allows for a contribution of cosmic rays from an extraordinary event that occurred ∼400 kyr ago. This type of event releases electrons/positrons with a total energy more than 10 times the average energy for usual supernovae, and its occurrence rate is lower than one three-hundredth of the usual supernova rate.

Evaluating diffusion dispersion across an extended range of b-values and frequencies: Exploiting gap-filled OGSE shapes, strong gradients, and spiral readouts.

PurposeTo address the long echo times and relatively weak diffusion sensitization that typically limit oscillating gradient spin‐echo (OGSE) experiments, an OGSE implementation combining spiral readouts, gap‐filled oscillating gradient shapes providing stronger diffusion encoding, and a high‐performance gradient system is developed here and utilized to investigate the tradeoff between b‐value and maximum OGSE frequency in measurements of diffusion dispersion (i.e., the frequency dependence of diffusivity) in the in vivo human brain. In addition, to assess the effects of the marginal flow sensitivity introduced by these OGSE waveforms, flow‐compensated variants are devised for experimental comparison.MethodsUsing DTI sequences, OGSE acquisitions were performed on three volunteers at b‐values of 300, 500, and 1000 s/mm2 and frequencies up to 125, 100, and 75 Hz, respectively; scans were performed for gap‐filled oscillating gradient shapes with and without flow sensitivity. Pulsed gradient spin‐echo DTI acquisitions were also performed at each b‐value. Upon reconstruction, mean diffusivity (MD) maps and maps of the diffusion dispersion rate were computed.ResultsThe power law diffusion dispersion model was found to fit best to MD measurements acquired at b = 1000 s/mm2 despite the associated reduction of the spectral range; this observation was consistent with Monte Carlo simulations. Furthermore, diffusion dispersion rates without flow sensitivity were slightly higher than flow‐sensitive measurements.ConclusionThe presented OGSE implementation provided an improved depiction of diffusion dispersion and demonstrated the advantages of measuring dispersion at higher b‐values rather than higher frequencies within the regimes employed in this study.

A class of integro-differential Fokker-Planck equations with space-dependent coefficient

Exact solution for the probability distribution function (PDF) in Laplace space is obtained for a class of integro-differential Fokker-Planck equations with different kernels and generic space-dependent diffusion coefficient. Besides, exact solutions for the PDF and n-moment are also obtained for different kernels and a power-law diffusion coefficient; they are analyzed and compared for different kernels.

Effect of Isothermal Aging and Copper Substrate Roughness on the SAC305 Solder Joint Intermetallic Layer Growth of High Temperature Storage (HTS)

This study aims to evaluate the effect of copper (Cu) substrate surface roughness on the intermetallic compound (IMC) growth and interfacial reaction of SAC305 lead-free solder joint after undergone an aging process. Aging process was conducted using high temperature storage (HTS) at temperature of 150 °C and aging times of 200, 400, 600, 800, and 1000 h. IMC morphology and growth were examined using infinite focus microscope (IFM). Then, the SAC305 solder joint IMC growth kinetic was measured based on power law relationship and diffusion coefficient formula. It was noted that the morphology of IMC for the rougher Cu substrate has scallop-shaped and uniform layer as compared to that of smoother Cu substrate for the initial exposure to the HTS. In addition, Cu substrate with Ra of 579 nm is the turning point for the creation of Cu6Sn5 towards more Cu3Sn of IMC. In addition, Cu substrate with Ra of 579 nm also acts as the turning point for the IMC growth of SAC305 solder joint on Cu substrate for the solid-state diffusion to be happened during 150 °C of aging from grain boundary dominant toward volume diffusion dominant.

Molecular Communication in Fractional Diffusive Channel

The molecular communication system with anomalous/fractional diffusion inside a one-dimensional (1-D) environment is considered. The time-dependent diffusivity is incorporated in terms of the power-law diffusivity, and the expression of first passage time density (FPTD) is derived. Further, the peak pulse time and peak concentration corresponding to the derived FPTD function are obtained. Moreover, the analysis is extended in terms of the average probability of error and throughput for the anomalous diffusion channel. The analytical results are validated through simulations.

A time-dependent diffusion MRI signature of axon caliber variations and beading

MRI provides a unique non-invasive window into the brain, yet is limited to millimeter resolution, orders of magnitude coarser than cell dimensions. Here, we show that diffusion MRI is sensitive to the micrometer-scale variations in axon caliber or pathological beading, by identifying a signature power-law diffusion time-dependence of the along-fiber diffusion coefficient. We observe this signature in human brain white matter and identify its origins by Monte Carlo simulations in realistic substrates from 3-dimensional electron microscopy of mouse corpus callosum. Simulations reveal that the time-dependence originates from axon caliber variation, rather than from mitochondria or axonal undulations. We report a decreased amplitude of time-dependence in multiple sclerosis lesions, illustrating the potential sensitivity of our method to axonal beading in a plethora of neurodegenerative disorders. This specificity to microstructure offers an exciting possibility of bridging across scales to image cellular-level pathology with a clinically feasible MRI technique.

Thermostatistics of overdamped motion with anisotropic drag forces

We investigate the consequences of anisotropic drag forces for the thermostatistics of systems of interacting particles in the overdamped motion regime. Systems of confined particles interacting via short-range forces and performing overdamped motion exhibit interesting thermodynamic features, and constitute useful models for various problems in physics and other areas. In particular, they proved useful for the study of interacting vortices in type-II superconductors. The dynamics of the above mentioned many-particle systems admits an effective description based on nonlinear Fokker-Planck equations with power-law diffusion terms. Most works studying this kind of systems, and their relationship with nonlinear Fokker-Planck equations, assume isotropic drag forces. In the present work we consider the effects of anisotropic drag forces on this kind of systems. We derive the corresponding nonlinear Fokker-Planck equations and investigate some of their main properties.

Corrosion and Mechanical Performance of Grade 92 Ferritic-Martensitic Steel After Exposure to Supercritical Carbon Dioxide

Grade 92 ferritic-martensitic steel is a candidate alloy for medium temperature (< 550 °C) components for the supercritical carbon dioxide (s-CO2) Brayton cycle. 1000 hours exposures were performed on base and welded material in s-CO2 at temperatures of 450 °C or 550 °C and compared to samples aged in Ar at 550 °C. Both s-CO2 exposures resulted in a duplex oxide growth and carburization, with 450 °C exhibiting carburization in a power law diffusion profile up to a depth of 200-250 µm, while 550 °C showed a linear profile up to a depth of 100 µm. The different profiles indicate much slower precipitation and coarsening of carbides at the lower temperature, allowing carbon to diffuse deeper into the material. However, 450 °C produced improved mechanical properties while 550 °C produced deteriorated properties. This was due to the higher density of carbon near the metal–oxide interface which leads to significant carbide coarsening and, subsequently, crack initiation and early failure. Additional exposure at 450 °C is predicted to increase deposited carbon, but further study would be needed to understand if and when carburization will produce a negative mechanical effect.

Global minimisers for anisotropic attractive–repulsive interactions

We prove the existence of global minimisers for a class of attractive–repulsive interaction potentials that are in general not radially symmetric. The global minimisers have compact support. For potentials including degenerate power-law diffusion, the interaction potential can be unbounded from below. Further, a formal calculation indicates that for non-symmetric potentials global minimisers may neither be radial symmetric nor unique.

Study of Brownian functionals for a Brownian process model of snow melt dynamics with purely time dependent drift and diffusion

In this paper, we investigate a Brownian motion (BM) with purely time dependent drift and diffusion by suggesting and examining several Brownian functionals, which characterize the stochastic model of water resources availability in snowmelt dominated regions with power law time dependent drift and diffusion. Snow melt process is modelled by a overdamped Langevin equation for a Brownian process with power law time dependent drift (μ(t) ~ qktα) and diffusion (D(t) ~ ktα) where they are proportional to each other. We introduce several probability distribution functions (PDFs) associated with such time dependent BMs. For instance, with initial starting value of snow amount H0, we derive analytical expressions for: (i) the PDF P(tf|H0) of the first passage time tf which specify the lifetime of such stochastic process, (ii) the PDF P(A|H0) of the area A till the first passage time and it provides us numerous valuable information about the average available water resources, (iii) the PDF P(M) associated with the maximum amount of available water M of the BM process before the complete melting of snow, and (iv) the joint PDF P(M;tm) of the maximum amount of available water M and its occurrence time tm before the first passage time. We further confirm our analytical predictions by computing the same PDFs with direct numerical simulations of the corresponding Langevin equation. We obtain a very good agreement of our theoretical predictions with the numerically simulated results. Finally, several nontrivial scaling behaviour in the asymptotic limits for the above mentioned PDFs are predicted, which can be verified further from experimental observation.

P-Laplace diffusion for distance function estimation, optimal transport approximation, and image enhancement

In this paper, we propose ADMM-based numerical schemes for power-law diffusion equations involving the p-Laplacian div(|∇u|p−2∇u) with constant (p=const) and variable (p=p(x)) exponents. Applications to distance function and optimal transport approximations (p is large and constant) and image enhancement (p is small and variable) are considered.

Ground states in the diffusion-dominated regime

We consider macroscopic descriptions of particles where repulsion is modelled by non-linear power-law diffusion and attraction by a homogeneous singular kernel leading to variants of the Keller–Segel model of chemotaxis. We analyse the regime in which diffusive forces are stronger than attraction between particles, known as the diffusion-dominated regime, and show that all stationary states of the system are radially symmetric non-increasing and compactly supported. The model can be formulated as a gradient flow of a free energy functional for which the overall convexity properties are not known. We show that global minimisers of the free energy always exist. Further, they are radially symmetric, compactly supported, uniformly bounded and C^infty inside their support. Global minimisers enjoy certain regularity properties if the diffusion is not too slow, and in this case, provide stationary states of the system. In one dimension, stationary states are characterised as optimisers of a functional inequality which establishes equivalence between global minimisers and stationary states, and allows to deduce uniqueness.

Wetting Front Analysis of the Richards’ Equation with Impervious Boundary

Abstract It is analyzed the wetting front of the Richards’ equation (RE) for horizontal infiltration problem with impervious layer in a finite medium, and obtained an approximate analytical solution by the series expansion technique. The present approximation is suitable for arbitrary diffusivity in RE and applied to simulate the changes of saturation after the front pass through the impervious point. Two examples about power law diffusivity are analyzed to confirm the accuracy of present solution.