One-dimensional Diffusion Research Articles

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

We present a comparison of Physics Informed Neural Networks (PINN) and Variational Physics Informed Neural Networks (VPINN) with higher-order and continuity Finite Element Method (FEM). We focus on the one-dimensional advection-dominated diffusion problem and the two-dimensional Eriksson–Johnson model problem. We show that the standard Galerkin method for FEM cannot solve this problem on uniform grid. We discuss the stabilization of the advection-dominated diffusion problem with the Petrov–Galerkin (PG) formulation and present the FEM solution obtained with the PG method. The main benefit of using a stabilization method is that it can deliver a good-quality approximation to the solution on a mesh that is not fully refined towards the singularity. We employ PINN and VPINN methods, defining several strong and weak loss functions. We compare the training and solutions of PINN and VPINN methods with higher-order FEM methods. We consider a case with uniform FEM and uniform distribution of points for PINN, as well as uniform distribution of test functions for VPINN. We also consider adaptive FEM, refined towards edge singularity, and non-uniform distribution of points for PINN, as well as non-uniform distribution of test functions for VPINN.

Short-time expansion of one-dimensional Fokker-Planck equationswith heterogeneous diffusion.

We formulate a short-time expansion for one-dimensional Fokker-Planck equationswith spatially dependent diffusion coefficients, derived from stochastic processes with Gaussian white noise, for general values of the discretization parameter 0≤α≤1 of the stochastic integral. The kernel of the Fokker-Planck equation(the propagator) can be expressed as a product of a singular and a regular term. While the singular term can be given in closed form, the regular term can be computed from a Taylor expansion whose coefficients obey simple ordinary differential equations. We illustrate the application of our approach with examples taken from statistical physics and biophysics. Furthermore, we show how our formalism allows us to define a class of stochastic equationswhich can be treated exactly. The convergence of the expansion cannot be guaranteed independently from the discretization parameter α.

Spatial diffusion of potentially toxic elements in soils around non-ferrous metal mines

This study focuses on the diffusion patterns of principal ore-forming elements (Pb and Zn) and associated elements (Cd, Cu, Cr, and As) in lead-zinc ore. Sampling points in upwind and downwind directions of lead-zinc ore areas at various densities (1 N/km2 – 4 N/km2) were categorized. This study analyzed the statistical relationship between the content of PTEs in the soil around lead-zinc ore and the source strength and dominant wind direction, constructed one-dimensional and two-dimensional diffusion model, and simulated the EER scope caused by PTEs. The findings indicate that: (1) concerning source strength, the content of PTEs in soils of high-density ore aggregation areas is significantly higher than in low-density ore aggregation areas. However, the impact of source strength decreases with decreasing ore grade, with a difference in Pb content of 1.71 times among principal ore-forming elements and almost consistent Cd content among associated elements. (2) Regarding the transport pathways, for most PTEs, the inverse proportion coefficients downwind are higher than upwind, approximately 1.18–3.63 times, indicating greater migration distances of PTEs downwind due to atmospheric dispersion. (3) By establishing a two-dimensional risk diffusion model, the study simulates the maximum radius of risk diffusion (r = 5.7 km), the 50% probability radius (r = 3.1 km), and the minimum radius (r = 0.8 km) based on the maximum, median, and minimum values statistically obtained from the EER. This study provides a scientific basis for implementing preventive measures for PTEs accumulation in soil within different pollution ranges. Different risk prevention and control measures should be adopted for PTEs accumulation in soil within the three ranges after cutting off pollution sources. Subsequent research should further investigate the impact and contribution of atmospheric transmission and surface runoff on the diffusion of PTEs in areas with high risk near lead-zinc ore.

LaCl3-based sodium halide solid electrolytes with high ionic conductivity for all-solid-state batteries

To enable high performance of all solid-state batteries, a catholyte should demonstrate high ionic conductivity, good compressibility and oxidative stability. Here, a LaCl3-based Na+ superionic conductor (Na1−xZrxLa1−xCl4) with high ionic conductivity of 2.9 × 10−4 S cm−1 (30 °C), good compressibility and high oxidative potential (3.80 V vs. Na2Sn) is prepared via solid state reaction combining mechanochemical method. X-ray diffraction reveals a hexagonal structure (P63/m) of Na1−xZrxLa1−xCl4, with Na+ ions forming a one-dimensional diffusion channel along the c-axis. First-principle calculations combining with X-ray absorption fine structure characterization etc. reveal that the ionic conductivity of Na1−xZrxLa1−xCl4 is mainly determined by the size of Na+-channels and the Na+/La3+ mixing in the one-dimensional diffusion channels. When applied as a catholyte, the NaCrO2||Na0.7Zr0.3La0.7Cl4||Na3PS4||Na2Sn all-solid-state batteries demonstrate an initial capacity of 114 mA h g−1 and 88% retention after 70 cycles at 0.3 C. In addition, a high capacity of 94 mA h g−1 can be maintained at 1 C current density.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

Escape from textured adsorbing surfaces.

The escape dynamics of sticky particles from textured surfaces is poorly understood despite importance to various scientific and technological domains. In this work, we address this challenge by investigating the escape time of adsorbates from prevalent surface topographies, including holes/pits, pillars, and grooves. Analytical expressions for the probability density function and the mean of the escape time are derived. A particularly interesting scenario is that of very deep and narrow confining spaces within the surface. In this case, the joint effect of the entrapment and stickiness prolongs the escape time, resulting in an effective desorption rate that is dramatically lower than that of the untextured surface. This rate is shown to abide a universal scaling law, which couples the equilibrium constants of adsorption with the relevant confining length scales. While our results are analytical and exact, we also present an approximation for deep and narrow cavities based on an effective description of one-dimensional diffusion that is punctuated by motionless adsorption events. This simple and physically motivated approximation provides high-accuracy predictions within its range of validity and works relatively well even for cavities of intermediate depth. All theoretical results are corroborated with extensive Monte Carlo simulations.

Coupled thermomechanical analysis using isoparametric curved shell elements

This study focuses on the coupled thermo-mechanical formulation in isoparametric shell elements and its implementation in curved laminar structures. The formulation is based on the direct isoparametric formulation, specifically designed for flat and curved shell elements, and relies on the theory of degenerated solids. The research provides a comprehensive description of the coupled thermo-mechanical analysis, including the mechanical equilibrium equations, the interrelations between stress, temperature, displacement, deformation, and the conservation of thermal energy. It adopts a simultaneous approach to addressing the impacts of temperature and mechanical loads. The paper elaborates on various numerical examples that substantiate the formulation. These examples include simulations that encompass linear scenarios, which are then compared to analytical solutions and results derived from other numerical software. The examples highlighted include one-dimensional temperature diffusion, radial diffusion, and the thermal behavior of a cylindrical cover and a pipe under a linear temperature gradient. These simulations demonstrate the formulation's capacity to accurately capture the interactions between temperature variations and structural displacements. They also confirm the alignment of the proposed model with existing analytical and numerical solutions. In conclusion, the research provides a good framework for coupled thermo-mechanical analysis. The flexibility of the formulation is evidenced across different configurations and scenarios, accommodating various types of boundary conditions such as constant and linear temperature fields, distributed loads, and constraints on displacements and rotations. It effectively manages temperature flows across one, two, and three dimensions, highlighting its extensive applicability in the realm of structural analysis. This versatility underscores its broad applicability in the field of structural analysis.

Investigation on the mechanism of self-healing effect of cementitious capillary crystalline waterproofing material on concrete and enhancement of its resistance to chloride erosion

To enhance the compactness of concrete in coastal areas and impede the ingress of chloride solutions into the structures, Cementitious capillary crystalline waterproofing material (CCCW) was internal incorporated into concrete. This study employed one-dimensional chloride immersion diffusion tests and wet-dry cycle tests to emulate the deterioration of concrete under natural conditions due to chloride ions. The optimal dosage of CCCW was determined by conducting chemical titration to measure the chloride ion concentration at various depths within the specimens. Subsequently, the influence of incorporating CCCW on the compressive strength of concrete under chloride conditions was assessed. Finally, employing experimental methods such as MIP, SEM and EDS, the pores within cementitious materials and the formation of internal crystals was examined at a microscopic level. Research indicates that in chloride salt environments, the penetration of salt solutions can trigger the formation of spider silk biomimetic shape crystalline structures within the internal pores of CCCW. thereby enhancing the compactness of concrete to achieve a self-healing effect while preventing the intrusion of chloride salt solutions.

The surface temperatures of flat areas on the Moon

Lunar surface temperature (LST) is a quantity of special interest for interpreting the thermal character of the regolith. It is affected by the solar irradiance, earthshine, and heat flow for the flat areas on the Moon. We present an improved transient temperature model to calculate temperatures of lunar flat surfaces. The model consists of one-dimensional thermal diffusion equation and two boundary conditions. The improved lunar surface boundary condition allows one to precisely determine the effective solar irradiance (ESI) and earthshine. The simulated surface temperatures suggest consistency with the measured temperatures from the thermocouples of Apollo 15 and 17 heat flow experiments. From the LST simulated with the improved model, it is found the annual-seasonal variations present obvious latitude characters, with the highest surface temperature occurring in late October, November, and December separately at high (>∼72°) latitudes, middle latitudes, and low (<∼20°) latitudes, respectively; the lowest surface temperatures occur in late July. Furthermore, the discrepancies between the maximum and minimum temperatures decrease as latitude increases, as do the maximum and minimum lunar surface temperatures. The surface daytime temperature would change by 179.4 K with a 1322.5 W/m2 change in the ESI. A 0.12 W/m2 and 0.02 W/m2 change of the earthshine and heat flow would lead to 0.5 and 0.09 K in surface nighttime temperature, respectively.

Concentration of quasi-stationary distributions for one-dimensional diffusions with applications

Concentration of quasi-stationary distributions for one-dimensional diffusions with applications

Solution of Fishers equation using homotopy perturbation Aboodh transform method

In this study, a nonlinear one-dimensional reaction Diffusion Equation called Fisher’s equation is solved using Aboodh Transform coupled with HPM. This method is applied to the general one-dimensional Fishers equation. This technique is demonstrated with three unique examples with different initial conditions solved with the values of β=1 and a=0. This simplicity of Aboodh transform makes the rapid rate of convergence from approximate to exact solution answers makes it easy and dependable.

RNase E searches for cleavage sites in RNA by linear diffusion: direct evidence from single-molecule FRET.

The ability of obstacles in cellular transcripts to protect downstream but not upstream sites en masse from attack by RNase E has prompted the hypothesis that this mRNA-degrading endonuclease may scan 5'-monophosphorylated RNA linearly for cleavage sites, starting at the 5' end. However, despite its proposed regulatory importance, the migration of RNase E on RNA has never been directly observed. We have now used single-molecule FRET to monitor the dynamics of this homotetrameric enzyme on RNA. Our findings reveal that RNase E slides along unpaired regions of RNA without consuming a molecular source of energy such as ATP and that its forward progress can be impeded when it encounters a large structural discontinuity. This movement, which is bidirectional, occurs in discrete steps of variable length and requires an RNA ligand much longer than needed to occupy a single RNase E subunit. These results indicate that RNase E scans for cleavage sites by one-dimensional diffusion and suggest a possible molecular mechanism.

The role of nucleotide opening dynamics in facilitated target search by DNA-repair proteins

Preserving the genomic integrity stands a fundamental necessity, primarily achieved by the DNA repair proteins through their continuous patrolling on the DNA in search of lesions. However, comprehending how even a single base-pair lesion can be swiftly and specifically recognized amidst millions of base-pair sites remains a formidable challenge. In this study, we employ extensive molecular dynamics simulations using an appropriately tuned model of both protein and DNA to probe the underlying molecular principles. Our findings reveal that the dynamics of a non-canonical base generate an entropic signal that guides the one-dimensional search of a repair protein, thereby facilitating the recognition of the lesion site. The width of the funnel perfectly aligns with the one-dimensional diffusion length of DNA-binding proteins. The generic mechanism provides a physical basis for rapid recognition and specificity of DNA damage sensing and recognition.

A deconvolution analysis strategy on the thermoreconstitution process of Co/Fe layered double hydroxides: Complementary correlations between physicochemical variations and thermokinetic analyses

A variety of physicochemical analyses were systematically conducted on the complex thermoreconstitution process of Co/Fe layered double hydroxides, and deconvolution analysis strategy was utilized in elucidating the reconstitution process. The thermal analysis of Co/Fe layered double hydroxides indicated that approximately 27.8 % weight loss, three weight loss peaks and two pronounced endothermic phenomena were detected via thermogravimetry, differential thermogravimetry, and differential scanning calorimetry, respectively. In the thermoreconstitution process of Co/Fe layered double hydroxides, scanning electron microscopy images indicated that the layered structure gradually collapsed, fractured into pieces, and partially melted. Meanwhile, significant X-ray diffraction peak broadening phenomena were detected and further confirmed by the d-spacing values in high-resolution transmission electron microscope images, which were mainly caused by the lattice expansion effect and the heterogeneous microstrain effect. Three reactions were deconvolved from the differential thermogravimetry curves, and the thermokinetic models of Reaction 2 indicated that volatile components in the form of H2O and CO2 escaped from the solid-state residuals via a one-dimensional diffusion reaction model. The major products in solid-state residuals were cubic Cobalt(III) oxide and Iron(II, III) oxide. The thermokinetic models played a complementary role in elucidating the physicochemical variations that occurred throughout the thermoreconstitution process of Co/Fe layered double hydroxides.

Pyrolysis characteristics, kinetics and products of flexible polyurethane foam in luxury cruise ship via TG, FTIR, GC-MS and Shuffled Complex Evolution

Flexible polyurethane foam is widely used in furniture upholstery, car seats, and other products. Its safety has attracted increasing attention, especially fire safety in luxury cruise ship. Thermogravimetry-Fourier Transform Infrared Spectroscopy-Gas Chromatography-Mass Spectrometry (TG-FTIR-GC-MS) is used to explore the pyrolysis characteristics of flexible polyurethane foam. The experimental results show that three weight loss peaks can be observed in the pyrolysis of flexible polyurethane foam under nitrogen atmosphere, which implies that the pyrolysis behaviors of isocyanate, polyol and catalyst are responsible for the peaks. The pyrolysis process of flexible polyurethane foam can be divided into two regions: the fifth-order reaction (F5) model and the one-dimensional diffusion (D1) model corresponding to regions I and II, respectively. The main pyrolysis intermediate function groups includes CO2, gaseous H2O, C-O, CO, C-H and other functional group stretching products. The amount of intermediate function groups generated in order of most to least is C-O > C-H > CO2 > CO > H2O. Six main products are generated under the N2 atmosphere, including Ethane, 2-Octen-1-ol, 3,7-dimethyl-isobutyrate, Diisopropyl ether, 1-Propoxypropan-2-yl acetate, Z-8-Methyl-9-tetradecenoic acid, Methyl 7,9-tridecadienyl ether and N-(2-Methylbutyl) undeca-(2E,4E)-diene-8,10-diynamide. The Shuffled Complex Evolution (SCE) algorithm is used to optimize the experimental pyrolysis kinetic parameters, and the optimized data show good agreement with the experimental data.

A diffusion model for liquid metal dealloying. Application to NiCu precursors dealloyed in liquid Ag

Liquid metal dealloying is a promising technique to elaborate porous metallic materials with potential applications for catalysis and energy storage. It consists in the selective dissolution of an element out of a precursor alloy by immersing it in a liquid bath. To model this process, we introduce a one-dimensional diffusion model that relies on the thermodynamic properties of the ternary system, and on a maximum velocity criterion of the reaction front. The model is compared to experimental measurements conducted on NiCu samples dealloyed in liquid Ag. Regarding the kinetics, the model enables to reach experimental time-scales, replicate experimental trends, and reveal the influence of the ligament structure on the dealloying rate. Also, by assuming thermodynamic equilibria between solid and liquid phases in the dealloyed region, the microstructure gradient obtained numerically is in satisfactory agreement with experimental measurements. These numerical results are obtained without adjustable parameters, which opens the possibility to predict dealloyed microstructures as function of the process parameters (precursor and bath composition, dealloying time and temperature, etc.).

Optimal finite-differences discretization for the diffusion equation from the perspective of large-deviation theory

When applying the finite-differences method to numerically solve the one-dimensional diffusion equation, one must choose discretization steps Δx, Δt in space and time, respectively. By applying large-deviation theory on the discretized dynamics, we analyze the numerical errors due to the discretization, and find that the (relative) errors are especially large in regions of space where the concentration of particles is very small. We find that the choice Δt=Δx2/(6D) , where D is the diffusion coefficient, gives optimal accuracy compared to any other choice (including, in particular, the limit Δt→0 ), thus reproducing the known result that may be obtained using truncation error analysis. In addition, we give quantitative estimates for the dynamical lengthscale that describes the size of the spatial region in which the numerical solution is accurate, and study its dependence on the discretization parameters. We then turn to study the advection–diffusion equation, and obtain explicit expressions for the optimal Δt and other parameters of the finite-differences scheme, in terms of Δx, D and the advection velocity. We apply these results to study large deviations of the area swept by a diffusing particle in one dimension, trapped by an external potential ∼|x| . We extend our analysis to higher dimensions by combining our results from the one dimensional case with the locally one-dimension method.

Elucidation of Mass Transport Phenomena in Highly Concentrated Electrolytes during Current Cycling Using In-Situ Interferometry and Finite Difference Method

Understanding electrolyte mass transfer during charge–discharge reactions is essential for developing next-generation storage batteries with high energy densities. In this study, we investigated Li+ transport in a highly concentrated electrolyte (HCE) consisting of an equimolar mixture of lithium bis(fluorosulfonyl)amide (LiFSA) and tetraglyme (G4) under current reversal and re-reversal. Concentration profiles of the electrolyte at a distance of 0–600 μm from the Li electrodes were obtained using in situ laser interferometry. The Li+ transference numbers and LiFSA diffusion coefficients were calculated from these profiles. Raman spectroscopy suggested that the coordination structure surrounding Li+ ions in the electrolytes mainly contributed to the transference number. A one-dimensional unsteady diffusion equation and the finite difference method were employed to simulate the concentration profiles. The maximum error percentage between the measured and simulated values was only 3%, confirming the accuracy and validity of the interferometric measurements. Our findings on Li-ion transfer in HCEs could promote the rational design of high-energy-density Li-ion batteries with higher cation transference numbers of electrolytes and charge–discharge rates.

Quasi-stationary distribution of a single species model under demographic stochasticity and Allee effects

In this paper, we utilize an absorbed diffusion process to model the dynamics of a single species under the influence of demographic stochasticity and component Allee effects. The trajectories of stochastic solutions exhibits multi-scale dynamics distinct from those of the corresponding mean-field model. The primary focus is on analyzing transient dynamics before extinction, which is described by the quasi-stationary distribution. The paper provides results on the existence, uniqueness, and exponential convergence to the quasi-stationary distribution for initial distributions with compact support. Due to the singularity of the noise at 0, we employ the idea of changing variables to transfer the model into one-dimensional Kolmogorov diffusions with a drift possibly exploding at 0, to prove the main results.

Asymptotic stability of planar rarefaction wave to a 2D hyperbolic-elliptic coupling system of the radiating gas on half-space

This paper studies the asymptotic stability of solutions to an initial-boundary value problem for a hyperbolic-elliptic coupling system on the two-dimensional half-space, where the data on the boundary and at the far field are prescribed as u_{-} and u_{+} , respectively. We show that the solution to the problem converges to the corresponding planar rarefaction wave for 0\le u_{-}<u_{+} as time tends to infinity. To the best of our knowledge, the stability results of planar rarefaction waves on half-space focus primarily on the single viscous conservation law because the rarefaction wave (one-dimensional diffusion wave) of the corresponding one-dimensional problem to scalar viscous conservation law is known. In other words, for a general high-dimensional system of equations, we cannot obtain the stability of planar rarefaction waves on half-space because we cannot construct the rarefaction wave of the corresponding one-dimensional problem. In this paper, we use the structure of the hyperbolic-elliptic coupling system to obtain the monotonic rarefaction wave of the corresponding one-dimensional hyperbolic-elliptic coupling system, and hence give the stability of the planar rarefaction wave on half-space. This can be viewed as the first result for the system of equations on the stability of planar rarefaction waves on half-space.