Derivation Of Diffusion Equation Research Articles

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.

Classical and Quantum Models of Diffusion

The objective of the paper is twofold: first, to review the classical diffusion models and show the approximations at the origin of the parabolic character of the classical equations; second, to demonstrate a connection between the quantum and classical models of diffusion. As diffusion is inherently related to the motion of constituents, the consistent models are framed within the dynamics of mixtures. The derivation of diffusion equations is then determined based on the related, pertinent approximations.

Boundary conditions at a thin membrane for normal diffusion, classical subdiffusion, and slow subdiffusion processes

We consider three different diffusion processes in a system with a thin membrane: normal diffusion, classical subdiffusion, and slow subdiffusion. We conduct the considerations following the rule: If a diffusion equation is derived from a certain theoretical model, boundary conditions at a thin membrane should also be derived from this model with additional assumptions taking into account selective properties of the membrane. To derive diffusion equations and boundary conditions at a thin membrane, we use a particle random walk model in one‐dimensional membrane system in which space and time variables are discrete. Then we move from discrete to continuous variables. We show that the boundary conditions depend on both selective properties of the membrane and a type of diffusion in the system.

NUMERICAL INVESTIGATION OF THREE-DIMENSIONAL HAUSDORFF DERIVATIVE ANOMALOUS DIFFUSION MODEL

The Hausdorff derivative has been recognized as an efficient tool to characterize anomalous diffusion phenomena. This paper makes the first attempt to numerically investigate three-dimensional Hausdorff derivative diffusion equation by the method of fundamental solutions. The fundamental solution of the three-dimensional Hausdorff derivative diffusion equation is closely related to scaling transform and non-Euclidean Hausdorff fractal distance. The used method, as a meshless technique, is simple, accurate and efficient for solving the partial differential equations with fundamental solutions. Three numerical experiments have been conducted to reveal the effectiveness and rationality of Hausdorff derivative anomalous diffusion models with various temporal and spatial fractal orders, as well as the accuracy of the developed methodology.

Diffusion of Biological Organisms: Fickian and Fokker--Planck Type Diffusions

In this paper we derive diffusion equations in a heterogeneous environment. We consider a system of discrete kinetic equations that consists of two phenotypes of different turning frequencies. The ...

Extrinsic spin-charge coupling in diffusive superconducting systems

We present a theoretical study of diffusive superconducting systems with extrinsic spin-orbit coupling and arbitrarily strong impurity potential. We derive from a microscopic Hamiltonian a diffusion equation for the quasi-classical Green function, and demonstrate that all mechanisms related to the spin-orbit coupling are expressed in terms of three kinetic coefficients: the spin Hall angle, the spin current swapping coefficient, and the spin relaxation rate due to Elliott-Yafet mechanism. The derived diffusion equation contains a hitherto unknown term describing a spin-orbit torque that appears exclusively in the superconducting state. As an example, we provide a qualitative description of a magnetic vortex in a superconductor with triplet correlations, and show that the novel term describes a spin torque proportional to the vector product between the spectral angular momentum of the condensate and the triplet vector. Our equation opens up the possibility to explore spintronic effects in superconductors with no counterparts in the normal metallic state.

Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media

The anomalous diffusion in fractal isotropic/anisotropic porous media is characterized by the Hausdorff derivative diffusion model with the varying fractal orders representing the fractal structures in different directions. This paper presents a comprehensive understanding of the Hausdorff derivative diffusion model on the basis of the physical interpretation, the Hausdorff fractal distance and the fundamental solution. The concept of the Hausdorff fractal distance is introduced, which converges to the classical Euclidean distance with the varying orders tending to 1. The fundamental solution of the 3-D Hausdorff fractal derivative diffusion equation is proposed on the basis of the Hausdorff fractal distance. With the help of the properties of the Hausdorff derivative, the Huasdorff diffusion model is also found to be a kind of time-space dependent convection-diffusion equation underlying the anomalous diffusion behavior.

Point Transformations and Relationships among Linear Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

We present new connections among linear anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem (CLT). This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Anomalous Diffusion equations of O’Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we can tell) be expressed as examples of the O’Shaughnessy and Procaccia equations. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. We illustrate the ideas both analytically and with a detailed computational example for a non-trivial choice of point transformation. Finally, we summarize our results.

Thin-layer infrared drying characteristics of construction gypsum plaster and selection of a suitable drying model

In this study, infrared drying behaviour of construction gypsum plaster samples was investigated. Experimental studies were carried out by using an infrared dryer at different temperatures (50–85°C) and at atmospheric conditions. As a result, the falling rate period was the drying rate period for the drying of gypsum plaster samples at low temperatures studied (50°C and 65°C). But, three drying rate periods consisting of beginning, constant, and falling periods occurred at the high temperature studied (85°C). FTIR and SEM analyses were performed to determine microstructural and surface properties of raw material and dried gypsum samples. A typical FTIR spectra for construction gypsum plaster samples have been obtained. It has been observed that dried gypsum plasters at the temperatures of 50°C and 65°C have a porous structure. On the contrary, raw material and dried gypsum at temperature of 85°C were no almost porous structure according to the gypsum plasters at other temperatures studied. Thus, it may be said that the drying process for construction gypsum plaster sample should be performed at temperatures below 85°C. In this work, the experimental drying data have also been fitted to various model equations by statistical testing. The most suitable model equations for drying of construction gypsum plasters have been determined by analysing the statistical test parameters such as chi-square, RMSE, and RSS at the temperatures studied. It has been founded that these models are Page and Diffusion equations. Also, derived forms of Page and Diffusion equations have been used to obtain an equation containing both drying time and temperature by multiple nonlinear regression analysis. As a result, it has been determined that a model derived Diffusion equation is the best representing the drying of CGP. Finally, the effective diffusivities and activation energy for the drying process of construction gypsum plaster samples have been calculated. The effective diffusivity values were 2.9920×10−9m2/s, 3.7527×10−9m2/s, and 3.9386×10−9m2/s at temperatures of 50°C, 65°C, and 85°C, respectively. The activation energy value was also 7.3795kJ/mol.

A non-local structural derivative model for characterization of ultraslow diffusion in dense colloids

Ultraslow diffusion has been observed in numerous complicated systems. Its mean squared displacement (MSD) is not a power law function of time, but instead a logarithmic function, and in some cases grows even more slowly than the logarithmic rate. The distributed-order fractional diffusion equation model simply does not work for the general ultraslow diffusion. Recent study has used the local structural derivative to describe ultraslow diffusion dynamics by using the inverse Mittag–Leffler function as the structural function, in which the MSD is a function of inverse Mittag–Leffler function. In this study, a new stretched logarithmic diffusion law and its underlying non-local structural derivative diffusion model are proposed to characterize the ultraslow diffusion in aging dense colloidal glass at both the short and long waiting times. It is observed that the aging dynamics of dense colloids is a class of the stretched logarithmic ultraslow diffusion processes. Compared with the power, the logarithmic, and the inverse Mittag–Leffler diffusion laws, the stretched logarithmic diffusion law has better precision in fitting the MSD of the colloidal particles at high densities. The corresponding non-local structural derivative diffusion equation manifests clear physical mechanism, and its structural function is equivalent to the first-order derivative of the MSD.

Fractional and fractal derivative models for transient anomalous diffusion: Model comparison

Transient anomalous diffusion characterized by transition between diffusive states (i.e., sub-diffusion and normal-diffusion) is not uncommon in real-world geologic media, due to the spatiotemporal variation of multiple physical, hydrologic, and chemical factors that can trigger non-Fickian diffusion. There are four fractional and fractal derivative models that can describe transient diffusion, including the distributed-order fractional diffusion equation (D-FDE), the tempered fractional diffusion equation (T-FDE), the variable-order fractional diffusion equation (V-FDE), and the variable-order fractal derivative diffusion equation (H-FDE). This study evaluates these models for transient sub-diffusion by comparing their mean squared displacement (which is the criteria for diffusion state), breakthrough curves (exhibiting nuance in diffusive state transition), and possible hydrogeologic origin (to build a potential link to medium properties). Results show that the T-FDE captures the slowest transition from sub-diffusion to normal-diffusion, and the D-FDE model only captures transient diffusion ending with sub-diffusion. The other two models, V-FDE and H-FDE, define a time-dependent scaling index to characterize complex transition states and rates. Preliminary field application shows that the V-FDE model, which provides a flexible transition rate, is appropriate to capture the fast transition from sub-diffusion to normal-diffusion for transport of a fluorescent water tracer dye (uranine) through a small-scale fractured aquifer. Further evaluations are needed using field measurements, so that practitioners can select the most reliable model for real-world applications.

A spatial structural derivative model for ultraslow diffusion

This study investigates the ultraslow diffusion by a spatial structural derivative, in which the exponential function ex is selected as the structural function to construct the local structural derivative diffusion equation model. The analytical solution of the diffusion equation is a form of Biexponential distribution. Its corresponding mean squared displacement is numerically calculated, and increases more slowly than the logarithmic function of time. The local structural derivative diffusion equation with the structural function ex in space is an alternative physical and mathematical modeling model to characterize a kind of ultraslow diffusion.

Oxygen diffusion in hcp metals from first principles

Oxygen interstitial site energies and migration barriers in 15 hexagonal close-packed (hcp) metals have been calculated with first-principles density functional theory. Multiple hcp systems show a preference for the hexahedral site over the tetrahedral site, as well as a stable crowdion site. More surprisingly, in more than half of the hcp systems, the oxygen does not choose the large octahedral interstitial as its ground state. We explain this result based on the effective valence of the metal from crystal-field splitting and the $c/a$ ratio. Diffusion constants for oxygen in all 15 hcp systems are calculated from analytically derived diffusion equations and match available experimental data.

Curl flux induced drift in stochastic differential equations in the zero-mass limit

Abstract We consider the nonlinear stochastic dynamics of dissipative Hamiltonian systems with state-dependent friction and diffusion connected by the fluctuation–dissipation relation in high dimensions. The system under study has a close connection to Ao’s framework in constructing a dynamical potential for non-equilibrium processes without detailed balance. We study the limiting case where the mass approaches zero and give a new and complete derivation of effective stochastic differential equations. Using the Ito stochastic integral convention, we show that the limiting effective Langevin equations have a new drift term. This extra term happens to be identical to the corresponding anti-Ito (or isothermal) integral (requiring constant temperature) in one dimension. We, however, cannot obtain this additional drift term using conventional stochastic integrals in high dimension. It is interesting to note that in a high-dimensional system, a curl flux induced drift may appear even if the diffusion matrix is constant. Our findings are supported by numerical simulations. We further analyze and discuss the role of this new drift term in calculating the classic escape time. For the first time, to our knowledge, the relation between the escape rate and the anti-Ito integral is presented. We also demonstrate that the derived diffusion equations give a new sampling algorithm which can increase convergence speed in a simple two-dimensional example.

Effect of local velocity on diffusion-induced stress in large-deformation electrodes of lithium-ion batteries

In this work, the contribution of local velocity to the resultant flux of lithium in lithium-ion battery is introduced into the diffusion equation to describe the migration of lithium in the active material of electrodes. The effect of the local velocity on the stress evolution in a spherical electrode made of silicon is analyzed, using the derived diffusion equation and nonlinear theory of elasticity. Two boundary conditions at the surface of the electrode, which represent two extreme conditions of real electrode materials, are used in the stress analysis: one is stress-free, and the other is immobile. The numerical results with the stress-free boundary condition suggest that the effect of the local velocity on the distribution of radial stress and hoop stress increases with the increase of time and the effect of the local velocity on the distribution of lithium is relatively small. In comparison with the results without the effect of the local velocity, the effect of the local velocity is negligible for the immobile boundary condition. The numerical result shows that the use of the immobile boundary condition leads to the decrease of von-Mises stress, which likely will retard the mechanical degradation of electrode and improve the electrochemical performance of lithium-ion battery.

Effect of local deformation on the coupling between diffusion and stress in lithium-ion battery

It is known that the local deformation in electrode plays an important role in controlling the migration rate of lithium in lithium-ion battery, which can alter the stress state in electrode. This work derives a new diffusion equation, which takes account of the effect of the local velocity in an electrode, to describe the migration of lithium in the electrode. The effects of the local velocity on the lithium migration and stress evolution are analyzed, using the derived diffusion equation and the theory of linear elasticity. Numerical analysis of diffusion-induced stress in an isotropic, spherical electrode is performed. The numerical results show that there is a larger concentration gradient in the spherical electrode with the effect of local velocity than that without the effect of local velocity, which results in larger magnitudes of radial stress, hoop stress, and von-Mises stress than the corresponding magnitudes without the effect of local velocity. The effects of the current density on the distributions of the lithium concentration and stresses are also analyzed. For a small lithiation rate, the concentration gradient is relatively small, and the stresses in the spherical electrode are small. The results suggest that using small lithiation rates can retard the electrode from fast capacity fading and mechanical degradation.

2D Multitransient Electromagnetic Response Modeling of South China Shale Gas Earth Model Using an Approximation of Finite Difference Time Domain with Uniaxial Perfectly Matched Layer

In this study, we introduced multitransient electromagnetic (MTEM) method as an effective tool for shale gas exploration. We combined the uniaxial perfectly matched layer (UPML) equation with the first derivative diffusion equation to solve for a finite difference time domain (FDTD) UPML equation, which was discretized to form an algorithm for 3D modeling of earth impulse response and used in modeling MTEM response over 2D South China shale gas model. We started with stepwise demonstration of the UPML and the FDTD algorithm as an effective tool. Subsequently, quantitative study on the convergence of MTEM earth impulse response was performed using different grid setup over a uniform earth material. This illustrates that accurate results can be obtained for specified range of offset. Furthermore, synthetic responses were generated for a set of geological scenarios. Lastly, the FDTD algorithm was used to model the MTEM response over a 2D shale gas earth model from South China using a PRBS source. The obtained apparent resistivity section from the MTEM response showed a similar geological setup with the modeled 2D South China shale gas section. This study confirmed the competence of MTEM method as an effective tool for unconventional shale gas prospecting and exploitation.

Anomalous diffusion: fractional derivative equation models and applications in environmental flows

Heterogeneity embedded in natural media and flow field challenge the application of Ficks 1st Law in anomalous diffusion well documented in many disciplines. Anomalous diffusion is one of the major topics in theoretical physics and statistical mechanics, and it is also the fundamental physical process with good potential application in environmental and hydrologic sciences and engineering. As a novel modeling tool in mathematics and physics, the fractional-order derivative diffusion equation models characterize anomalous diffusion with history-dependence and spatial non-locality, accurately describe the tailing in breakthrough curves of solute transport. We summarize the recent progresses and discuss the key challenges of fractional derivative diffusion equation models including the existed research and current development, fractional derivative modeling, numerical algorithms, and related applications in the field of environmental fluid mechanics. Here also made some preliminary discussions on issues of fractional derivative diffusion equation model, such as statistical description, model parameter determination and dimensional analysis, which may contribute to the further study of anomalous diffusion.

Fractional derivative anomalous diffusion equation modeling prime number distribution

Abstract This study suggests that the power law decay of prime number distribution can be considered a sub-diffusion process, one of typical anomalous diffusions, and could be described by the fractional derivative equation model, whose solution is the statistical density function of Mittag-Leffler distribution. It is observed that the Mittag-Leffler distribution of the fractional derivative diffusion equation agrees well with the prime number distribution and performs far better than the prime number theory. Compared with the Riemann’s method, the fractional diffusion model is less accurate but has clear physical significance and appears more stable. We also find that the Shannon entropies of the Riemann’s description and the fractional diffusion models are both very close to the original entropy of prime numbers. The proposed model appears an attractive physical description of the power law decay of prime number distribution and opens a new methodology in this field.

Study of spin transport in Dirac systems

We investigate spin Hall effect in a three-dimensional Dirac electron system. We derive diffusion equations in the Keldysh formalism and define a spin current from a spin diffusion equation. We solve the obtained diffusion equations in a steady state in the bulk and find that the bulk spin Hall conductivity is non-zero, though that for the conventional spin current is zero.