Classical Diffusion Research Articles

Recent Developments in Kramers' Theory of Reaction Rates.

In this short review, we provide an update of recent developments in Kramers' theory of reaction rates. After a brief introduction stressing the importance of this theory initially developed for chemical reactions, we briefly present the main theoretical formalism starting from the generalized Langevin equation and continue by showing the main points of the modern Pollak, Grabert and Hänggi theory. Kramers' theory is then sketched for quantum and classical surface diffusion. As an illustration the surface diffusion of Na atoms on a Cu(110) surface is discussed showing escape rates, jump distributions and diffusion coefficients as a function of reduced friction. Finally, some very recent applications of turnover theory to different fields such as nanoparticle levitation, microcavity polariton dynamics and simulation of reaction in liquids are presented. We end with several open problems and future challenges faced up by Kramers turnover theory.

Electron kinetics in a positive column of AC discharges in a dynamic regime

We have performed hybrid kinetic-fluid simulations of a positive column in alternating current (AC) argon discharges over a range of driving frequencies f and gas pressure p for the conditions when the spatial nonlocality of the electron energy distribution function (EEDF) is substantial. Our simulations confirmed that the most efficient conditions of plasma maintenance are observed in the dynamic regime when time modulations of mean electron energy (temperature) are substantial. The minimal values of the root mean square electric field and the electron temperature have been observed at f/p values of about 3 kHz Torr−1 in a tube of radius R = 1 cm. The ionization rate and plasma density reached maximal values under these conditions. The numerical solution of a kinetic equation allowed accounting for the kinetic effects associated with spatial and temporal nonlocality of the EEDF. Using the kinetic energy of electrons as an independent variable, we solved an anisotropic tensor diffusion equation in phase space. We clarified the role of different flux components during electron diffusion in phase space over surfaces of constant total energy. We have shown that the kinetic theory uncovers a more exciting and rich physics than the classical ambipolar diffusion (Schottky) model. Non-monotonic radial distributions of excitation rates, metastable densities, and plasma density have been observed in our simulations at pR > 6 Torr cm. The predicted off-axis plasma density peak in the dynamic regime has never been observed in experiments so far. We hope our results stimulate further experimental studies of the AC positive column. The kinetic analysis could help uncover new physics even for such a well-known plasma object as a positive column in noble gases.

From anomalous to classical diffusion in a nonlinear heat equation

From anomalous to classical diffusion in a nonlinear heat equation

Implementation, Validation and Study Case of a Landscape Evolution Model Algorithm

This paper presents a landscape evolution model based on physical processes – hillslope processes and fluvial erosion, transport, and deposition – solved by numerical methodology. That is, through the solution of differential equations approximated by numerical methods. In this case, hillslope processes are modeled through the classical diffusion equation, discretized by the finite volume method. Fluvial erosion, transport, and deposition are modeled by the fluvial potential equations (stream power law). For this, the approximation is performed by the finite difference method. The topography – initial condition – is set by digital elevation models, obtained from satellite images. These are Raster datasets, that each cell contains a representative elevation value. The drainage is determined through the classical algorithm D8, which performs a scan on the digital elevation model, tracing routes of greater slopes between the cells. The algorithm execution flowchart is presented, and the model is validated. Finally, a geomorphological study is presented in the Piratini river basin, showing thar developed model mimics largescale natural phenomena of watershed processes.

Traffic-Driven Epidemic Spreading in Networks: Considering the Transition of Infection From Being Mild to Severe.

Realistic epidemic spreading is usually driven by traffic flow in networks, which is not captured in classic diffusion models. Moreover, the progress of a node's infection from mild to severe phase has not been particularly addressed in previous epidemic modeling. To address these issues, we propose a novel traffic-driven epidemic spreading model by introducing a new epidemic state, that is, the severe state, which characterizes the serious infection of a node different from the initial mild infection. We derive the dynamic equations of our model with the tools of individual-based mean-field approximation and continuous-time Markov chain. We find that, besides infection and recovery rates, the epidemic threshold of our model is determined by the largest real eigenvalue of a communication frequency matrix we construct. Finally, we study how the epidemic spreading is influenced by representative distributions of infection control resources. In particular, we observe that the uniform and Weibull distributions of control resources, which have very close performance, are much better than the Pareto distribution in suppressing the epidemic spreading.

Inverse problems for fractional Schrödinger and subdiffusion equations

The inverse problems of determining the right-hand side of the Schrödinger and the sub-diffusion equations with the fractional derivative is considered. In the problem 1, the time-dependent source identification problem for the Schrödinger equation , in a Hilbert space is investigated. To solve this inverse problem, we take the additional condition with an arbitrary bounded linear functional . In the problem 2, we consider the subdiffusion equation with a fractional derivative of order , and take the аbstract operator as the elliptic part. The right-hand side of the equation has the form , where is a given function and the inverse problem of determining element is considered. The condition is taken as the over-determination condition, where is some interior point of the considering domain and is a given element. Obtained results are new even for classical diffusion equations. Existence and uniqueness theorems for the solutions to the problems under consideration are proved.

Fractional potential: A new perspective on the fractional Laplacian problem on bounded domains

In this work, the fractional Laplacian operator is studied based on its spectral definition for a bounded, one-dimensional domain. We first note that both the analytical and numerical solutions of the fractional diffusion equation subject to Neumann boundary conditions do not exist due to the need to raise the zero eigenvalue to a negative fractional index. To overcome this issue, we use the concept of regularisation and modify the Neumann boundary conditions to Robin type by introducing regularisation parameters that ensure the smallest eigenvalue of both the Sturm–Liouville system and the matrix representation of the Laplacian operator is nonzero. We then observe that these regularisation parameters can have a significant impact on the solution behaviour of the fractional diffusion model. This finding motivates the use of what we call the ‘fractional potential’ to define a new fractional model that is far less sensitive to the regularisation parameter and makes imposing fractional Neumann boundary conditions also feasible. Next, we compare the results of this new fractional model with a fractional Riesz problem subject to Neumann boundary conditions on a bounded symmetric domain. An interesting finding is that the steady-state solution for the fractional Laplacian problem leads to a constant function, while the fractional Riesz operator results in a function exhibiting boundary singularities. Finally, we propose a three-layered fractional model based on the Laplacian operator to naturally impose the traditional boundary conditions by using the idea of ‘geometric reconstruction’. We derive semi-analytical and numerical solutions for this model. An important outcome is that by choosing an extremely small scale for the boundary transition layer on both sides of the domain, this three-layered model can be used to approximate the fractional Laplacian problem on a bounded domain. The proposed model captures the fractional diffusion behaviour in the inner layer well while describing the classical diffusion behaviour near the boundary. We show that this model is ideal for modelling anomalous diffusion in a binary medium exhibiting distinct heterogeneity.

Topological Subordination in Quantum Mechanics

An example of non-Markovian quantum dynamics is considered in the framework of a geometrical (topological) subordination approach. The specific property of the model is that it coincides exactly with the fractional diffusion equation, which describes the geometric Brownian motion on combs. Both classical diffusion and quantum dynamics are described using the dilatation operator xddx. Two examples of geometrical subordinators are considered. The first one is the Gaussian function, which is due to the comb geometry. For the quantum consideration with a specific choice of the initial conditions, it corresponds to the integral representation of the Mittag–Leffler function by means of the subordination integral. The second subordinator is the Dirac delta function, which results from the memory kernels that define the fractional time derivatives in the fractional diffusion equation.

Fundamental solution of diffusion equationfor Kappa gas: Diffusion length for suprathermal electrons in solar wind.

A recent numerical treatment of data obtained by the Parker Solar Probe spacecraft describes the electron concentration in solar wind as a function of the heliocentric distance based on a Kappa distribution with spectral index κ=5. In this work, we derive and, subsequently, solve an entirely different class of nonlinear partial differential equationsdescribing the one-dimensional diffusion of a suprathermal gas. The theory is applied to describe the aforementioned data and we find a spectral index κ≳1.5 providing the widely acknowledged identification of Kappa electrons in solar wind. We also find that suprathermal effects increase the length scale of classical diffusion by one order of magnitude. Such a result does not depend on the microscopic details of the diffusion coefficient since our theory is based on a macroscopic formulation. Forthcoming extensions of our theory by including magnetic fields and relating our formulation to nonextensive statistics are briefly addressed.

On Construction of Partially Dimension-Reduced Approximations for Nonstationary Nonlocal Problems of a Parabolic Type

The main aim of this article is to propose an adaptive method to solve multidimensional parabolic problems with fractional power elliptic operators. The adaptivity technique is based on a very efficient method when the multidimensional problem is approximated by a partially dimension-reduced mathematical model. Then in the greater part of the domain, only one-dimensional problems are solved. For the first time such a technique is applied for problems with nonlocal diffusion operators. It is well known that, even for classical local diffusion operators, the averaged flux conjugation conditions become nonlocal. Efficient finite volume type discrete schemes are constructed and analysed. The stability and accuracy of obtained local discrete schemes is investigated. The results of computational experiments are presented and compared with theoretical results.

The Effects of Oxygen Dilution on Two-Stage Autoignition of Isolated N-Dodecane Droplets in Microgravity

ABSTRACT Two-stage autoignition of n-dodecane droplets with varying ambient oxygen concentrations in oxygen-nitrogen mixtures are investigated experimentally under microgravity conditions using high-speed shadowgraphy. The ambient pressure and temperature are held constant at 3 atm and 650 K, respectively, while the droplet initial diameter is fixed approximately at 1.2 mm. During the two-stage autoignition process, first a cool-flame front forms in the leaner regions farther away from the evaporating droplet, and it then propagates toward the fuel-rich region closer to the droplet surface, eventually encompassing the droplet. A hot-flame kernel is then initiated in the wake of the cool flame and very quickly expands, establishing a classical diffusion flame around the droplet. The first and second ignition delay times are measured from shadowgraphic images captured at 3000 frames per second. The first induction time is found to be insensitive to the ambient oxygen concentration, while the second induction time varies approximately as the negative 2 power of the oxygen mole fraction.

Transport of a passive scalar in wide channels with surface topography: An asymptotic theory

We generalize classical dispersion theory for a passive scalar to derive an asymptotic long-time convection–diffusion equation for a solute suspended in a wide, structured channel and subject to a steady low-Reynolds-number shear flow. Our asymptotic theory relies on a domain perturbation approach for small roughness amplitudes of the channel and holds for general surface shapes expandable as a Fourier series. We determine an anisotropic dispersion tensor, which depends on the characteristic wavelengths and amplitude of the surface structure. For surfaces whose corrugations are tilted with respect to the applied flow direction, we find that dispersion along the principal direction (i.e. the principal eigenvector of the dispersion tensor) is at an angle to the main flow direction and becomes enhanced relative to classical Taylor dispersion. In contrast, dispersion perpendicular to it can decrease compared to the short-time diffusivity of the particles. Furthermore, for an arbitrary surface shape represented in terms of a Fourier decomposition, we find that each Fourier mode contributes at leading order a linearly-independent correction to the classical Taylor dispersion diffusion tensor.

Evidence for the Self-organized Criticality Phenomenon in the Prompt Phase of Short Gamma-Ray Bursts

The prompt phase of gamma-ray bursts (GRBs) contains essential information regarding their physical nature and central engine, which are as yet unknown. In this paper, we investigate the self-organized criticality phenomenon in GRB prompt phases as done in X-ray flares of GRBs. We obtain the differential and cumulative distributions of 243 short GRB pulses, such as peak flux, FWHM, rise time, decay time, and peak time in the fourth BATSE Time-Tagged Event Catalog with the Markov Chain Monte Carlo technique. It is found that these distributions can be well described by power-law models. In particular, comparisons are made with 182 short GRB pulses in the third Swift GRB Catalog from 2004 December to 2019 July. The results are essentially consistent with the BATSE ones. We notice that there is no obvious power-law index evolution across different energy bands for either BATSE or Swift short GRBs. The joint analysis suggests that the GRB prompt phase can be explained by a fractal-diffusive self-organized criticality system with the spatial dimension S = 3 and the classical diffusion β = 1. Our findings show that GRB prompt phases and X-ray flares possess the very same magnetically dominated stochastic process and mechanism.

Diffusion phase-imaging in anisotropic media using non-linear gradients for diffusion encoding.

Diffusion MRI classically uses gradient fields that vary linearly in space to encode the diffusion of water molecules in the signal magnitude by tempering its intensity. In spin ensembles, a presumably equal number of particles move in positive and negative direction, resulting in approximately zero change in net phase. Hence, in classical diffusion weighted MRI with a linear gradient field, the phase does not carry any information as the incoherent motion of the spins only impacts the magnitude of the signal. Conversely, when the linear gradient field is replaced with one that varies quadratically over space, the diffusion of water molecules in anisotropic media does give rise to a change in net phase and preserves large portion of the signal around the saddle point of the gradient field. In this work, the phase evolution of anisotropic fibre phantoms in the presence of quadratic gradient fields was studied in Monte Carlo simulations and diffusion MRI experiments. The simulations confirm the dependence of the phase change on the degree of anisotropy of the media and the diffusion weighting, as predicted by the derived analytic model. First MR experiments show a phase change depending on the diffusion time in an anisotropic synthetic fibre phantom, and approximately zero phase change for the experiment repeated in an isotropic agar phantom. As predicted by the analytic model, an increase of the diffusion time by approximately a factor of two leads to an increase of approximately a factor of two in the signal phase.

NUMERICAL SIMULATION OF FRACTIONAL POWER DIFFUSION BIOSENSORS

The main aim of this paper is to propose new mathematical models for simulation of biosensors and to construct and investigate discrete methods for the efficient solution of the obtained systems of nonlinear PDEs. The classical linear diffusion operators are substituted with nonlocal fractional powers of elliptic operators. The splitting type finite volume scheme is used as a basic template for the introduction of new mathematical models. Therefore the accuracy of the splitting scheme is investigated and compared with the symmetric Crank-Nicolson scheme. The dependence of the approximation error on the regularity of the solution is investigated. Results of computational experiments for different values of fractional parameters are presented and analysed.

Linear Programming for Influence Maximization Problems in Social Networks Based on Diffusion Models

Influence maximization is a challenging problem that combines optimization and the study of social sciences. The purpose is to find the most influential individual within a society with a limited number of initial nodes. With the increasing use of social networks these years, Understanding and using forceful diffusion will spread social events or innovations far more widely than the conventional perspective of communication. Many studies have focused on providing algorithmic methods, ignoring the study of finding the optimal point. Although it is still difficult for computer community to settle massive influence maximization issues computationally, it presents a significant opportunity to build operations research-based models, approaches, and analyses at this interface. In the current research, the influence maximization problem is investigated as a discrete optimization issue on social networks. It focuses on the creation of precise techniques for resolving massive cases near the optimal and will provide two classical diffusion models describing the propagation of influence in networks. Finally, given the additional constraints considered, an extension of the trigger model with effective analysis is proposed.

Multi-spike solutions to the one-dimensional subcritical fractional Schnakenberg system

Singularly perturbed reaction–diffusion systems such as the Gierer–Meinhardt, Schankenberg, and Gray–Scott models are known to exhibit localized multi-spike solutions in which one of the species concentrates on a discrete collection of well-separated points. A recent focus has been the extension of the analysis, both rigorous and formal, of such localized solutions to the case where classical diffusion is replaced by anomalous diffusion exhibiting Lévy flights. Mathematically, such systems replace the classical Laplacian with the fractional Laplacian (−Δ)s where s is the fractional order. In this paper we consider the formal analysis of multi-spike solutions to the fractional Schnakenberg system with periodic boundary conditions when the fractional order of the activator satisfies 1/4<s1<1 and that of the substrate satisfies 1/2<s2<1. Using the method of matched asymptotic expansions we derive a nonlinear algebraic system that determines the structure of equilibrium solutions, a nonlocal eigenvalue problem that determines its linear stability over a fast, O(1), timescale, as well as a system of ordinary differential equations that determine the dynamics of multi-spike solutions over a slow, O(ɛ−2), timescale. We explicitly construct symmetric multi-spike solutions and derive stability criteria for instabilities arising over both fast and slow timescales. One of the key findings in this paper is that the slow timescale instability thresholds are, barring symmetry considerations, less degenerate than in the classical s2=1 case. In addition we numerically calculate bifurcation diagrams that show how asymmetric multi-spike solutions branch out from symmetric ones at distinct points for solutions with five or more spikes.

Exsolution Modeling and Control to Improve the Catalytic Activity of Nanostructured Electrodes.

In situ exsolution for nanoscale electrode design has attracted considerable attention because of its promising activity and high stability. However, fundamental research on the mechanisms underlying particle growth remains insufficient. Herein, cation-diffusion-determined exsolution is presented using an analytical model based on classical nucleation and diffusion. In the designed perovskite system, the exsolution trend for particle growth is consistent with this diffusion model, which strongly depends on the initial cation concentration and reduction conditions. Based on the experimental and theoretical results, a highly Ni-doped anode and an electrochemical switching technique are employed to promote exsolution and overcome growth limitations. The optimal cell exhibits an outstanding maximum power density of 1.7Wcm-2 at 900°C and shows no evident degradation when operating at 800°C for 240h under wet H2 . This study provides crucial insights into the developing and tuning of heterogeneous catalysts for energy-conversion applications.

An Application of the Distributed-Order Time- and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y-direction). In this work, we propose a distributed-order time- and space-fractional diffusion-wave equation to model a comb structure in the more general setting. The distributed-order time- and space-fractional diffusion-wave equation is firstly formulated to study the anomalous diffusion in the comb model subject to an irregular convex domain with the motivation that the time-fractional derivative considers the memory characteristic and the space one with the variable diffusion coefficient possesses the nonlocal characteristic. The finite element method is applied to obtain the numerical solution. The stability and convergence of the numerical discretization scheme are derived and analyzed. Two numerical examples of relevance to the comb model are given to verify the correctness of the numerical method. Moreover, the influence of the involved parameters on the three-dimensional and axial projection drawing particle distribution subject to an elliptical domain are analyzed, and the physical meanings are interpreted in detail.

When complex movement yields simple dispersal: behavioural heterogeneity, spatial spread and parasitism in groups of micro-wasps

BackgroundUnderstanding how behavioural dynamics, inter-individual variability and individual interactions scale-up to shape the spatial spread and dispersal of animal populations is a major challenge in ecology. For biocontrol agents, such as the microscopic Trichogramma parasitic wasps, an understanding of movement strategies is also critical to predict pest-suppression performance in the field.MethodsWe experimentally studied the spatial propagation of groups of parasitoids and their patterns of parasitism. We investigated whether population spread is density-dependent, how it is affected by the presence of hosts, and whether the spatial distribution of parasitism (dispersal kernel) can be predicted from the observed spread of individuals. Using a novel experimental device and high-throughput imaging techniques, we continuously tracked the spatial spread of groups of parasitoids over large temporal and spatial scales (8 h; and 6 m, ca. 12,000 body lengths). We could thus study how population density, the presence of hosts and their spatial distribution impacted the rate of population spread, the spatial distribution of individuals during population expansion, the overall rate of parasitism and the dispersal kernel (position of parasitism events).ResultsHigher population density accelerated population spread, but only transiently: the rate of spread reverted to low values after 4 h, in a “tortoise-hare” effect. Interestingly, the presence of hosts suppressed this transiency and permitted a sustained high rate of population spread. Importantly, we found that population spread did not obey classical diffusion, but involved dynamical switches between resident and explorer movement modes. Population distribution was therefore not Gaussian, though surprisingly the distribution of parasitism (dispersal kernel) was.ConclusionsEven homogenous asexual groups of insects develop behavioural heterogeneities over a few hours, and the latter control patterns of population spread. Behavioural switching between resident and explorer states determined population distribution, density-dependence and dispersal. A simple Gaussian dispersal kernel did not reflect classical diffusion, but rather the interplay of several non-linearities at individual level. These results highlight the need to take into account behaviour and inter-individual heterogeneity to understand population spread in animals.