Classical Diffusion Research Articles

Infective flooding in low-duty-cycle networks, properties and bounds

Flooding information is an important function in many networking applications. In some networks, as wireless sensor networks or some ad-hoc networks it is so essential as to dominate the performance of the entire system. Exploiting some recent results based on the distributed computation of the eigenvector centrality of nodes in the network graph and classical dynamic diffusion models on graphs, this paper derives a novel theoretical framework for efficient resource allocation to flood information in mesh networks with low duty-cycling without the need to build a distribution tree or any other distribution overlay. Furthermore, the method requires only local computations based on each node neighborhood. The model provides lower and upper stochastic bounds on the flooding delay averages on all possible sources with high probability. We show that the lower bound is very close to the theoretical optimum. A simulation-based implementation allows the study of specific topologies and graph models as well as scheduling heuristics and packet losses. Simulation experiments show that simple protocols based on our resource allocation strategy can easily achieve results that are very close to the theoretical minimum obtained building optimized overlays on the network.

Subdiffusive flow in a composite medium with a communicating (absorbing) interface

Two-dimensional subdiffusion in media separated by a partially communicating interface is considered. Starting with the appropriate Green’s functions, solutions are developed in terms of the Laplace transformation reflecting two circumstances at the interface: situations where there is perfect contact and situations where the interface offers a resistance. Asymptotic solutions are derived; limiting forms of the expressions reduce to known solutions for both classical diffusion and subdiffusion. Specifics are analyzed in depth with reference to flow in porous media with potential applications to the evaluation of the role of subsurface faults and flow in fractured rocks. Characteristics of the derivative responses are documented extensively as they are the linchpin for evaluation of pressure tests. Results given here may be used for evaluation at the Theis (1935; Eos Trans. AGU 2, 519–524) scale along with geological and geophysical properties, and production statistics. Yet a subdiffusive model does not imply a single value for properties. The method presented here may be extended to multiple contiguous media and to subdiffusive transport in many contexts (complex wellbores such as inclined, fractured and horizontal wells, situations such as sequestration, production of geothermal systems, etc.).

Anisotropic diffusion based on Fermi-Dirac distribution function and its application in the Shack-Hartman wavefront sensor

In this study, the anisotropic diffusion technique is applied to estimate spots in the noise signals of the Shack-Hartmann wavefront sensor. Based on the analysis of the classical anisotropic diffusion function and on an improved algorithm, a diffusion function is proposed based on the Fermi-Dirac distribution. It is proved mathematically that the new function has a higher convergence speed and a better performance. Monte Carlo simulations are used to verify the applicability of the new function subject to the noise limit and signal level. The simulation and experimental results show that the anisotropic diffusion algorithm can effectively filter out the noise. The integrity of the spots can be maintained, and the centroid detection accuracy and signal-to-noise ratio are also improved.

Time evolution law of Wigner operator in diffusion channel

As is well known, the evolution of quantum state can be replaced by its Wigner function’s time evolution. The Wigner function of a quantum state is the same as the density matrix of a quantum state, because they both contain many messages, such as the probability distribution and phases. Thus, the important information about the quantum state in the evolution process can be obtained more quickly and effectively by studying the Wigner function of a quantum state. In this paper, based on the classical diffusion equation, the diffusion equation of the quantum state density operator is derived by using the <i>P</i> representation of the density operator. Furthermore, by introducing the Weyl ordering symbol of the quantum operator, the corresponding Weyl quantization scheme is given. In addition, the evolution equation of Wigner operator in diffusion channel is established by using another phase space representation of density operator—Wigner function, and the solution form of Wigner operator is given. In this paper, we derive the evolution law of Wigner operator in quantum diffusion channel for the first time, that is, the form of Wigner operator at any time in the evolution process. Based on this conclusion, the evolution of coherent states through quantum diffusion channels is discussed.

Механизм формирования карбидокремниевого покрытия диффузионным методом в кремниевой засыпке на углерод-углеродных композиционных материалах

The pack cementation process, used for obtaining coatings, is described, as a rule, in terms of solid-phase diffusion, which leads to saturation of the near-surface layer of the workpiece with a certain element. This method allows the deposition of a silicon carbide coating on carbon-carbon composite materials, that increases its resistance to the action of high-temperature oxidizing environments. An analysis of the kinetic features of the formation of silicon carbide on a carbon material indicated the difference between this method and the classical diffusion saturation. It turned out that it is necessary to consider additional processes of gas formation that occurs in the powder filling. A mechanism has been proposed, according to which it is possible to transfer silicon through the gaseous reagent SiO, generated in the backfill with the help of oxide additives, like aluminum oxide, which were previously considered inert.

ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

Efficient calculation of the response statistics of two-dimensional fractional diffusive systems

Various natural phenomena are described by anomalous diffusion processes. Notable examples relate to the study of diffusion of tracers in particles in turbulent flows, or in the propagation of acoustic waves. In this context, the governing equations involve a fractional Laplacian operator, which replaces the classical Laplacian, and may involve nonlinear terms. This leads to problems described by fractional nonlinear diffusion equations. In general, no analytical solutions are available for determining the response of these systems. Thus, the development of approximate approaches circumventing the use of computationally demanding numerical techniques is desirable. This paper proposes a statistical linearization based approach, which allows calculating approximately, albeit iteratively, the response statistics. The method is developed using a recently proposed representation of the fractional Laplacian in conjunction with a mode expansion of the system response. It is implemented by introducing non-orthogonal eigenfunctions of the fractional Laplacian of the response, which are obtained from the linear modes of the classical diffusion equation. Such a representation allows deriving a system of nonlinear ordinary differential equations, which is linearized in a stochastic mean square sense. Then, the response statistics and power spectral density are determined by an iterative procedure. Numerical results pertaining to a system with white noise excitation demonstrate the efficiency of the proposed approximate approach. Further, comparisons with data from relevant Monte Carlo results assess the reliability of the estimated response.

Diffusion phenomena in a mixed phase space.

We show that, in strongly chaotic dynamical systems, the average particle velocity can be calculated analytically by consideration of Brownian dynamics in a phase space, the method of images, and the use of the classical diffusion equation. The method is demonstrated on the simplified Fermi-Ulam accelerator model, which has a mixed phase space with chaotic seas, invariant tori, and Kolmogorov-Arnold-Moser islands. The calculated average velocities agree well with numerical simulations and with an earlier empirical theory.

Theoretical analysis of particle size re-distribution due to Ostwald ripening in the fuel cell catalyst layer

AbstractThe limited durability of hydrogen fuel cells is one of the main obstacles in their wider adoption as a clean alternative technology for small scale electricity production. The Ostwald ripening of catalyst material is recognized as one of the main unavoidable degradation processes deteriorating the fuel cell performance and shortening its lifetime. The paper systematically studies how the modeling approach towards the electrochemically driven Ostwald ripening in the fuel cell catalyst differs from the classical diffusion driven models and highlights how these differences affect the resulting evolution of particle size distribution. At moderately low electric potential, root-law growth of mean particle size is observed with linear relation between mean particle size and standard deviation of particle size distribution, similar to Lifshitz-Slyozov-Wagner theory, but with broader and less skewed distribution. In case of high electric potential, rapid particle growth regime is observed and qualitatively described by redeposition of platinum from a highly oversaturated solution, revealing the deficiencies of the existing platinum degradation models at describing the Ostwald ripening in the fuel cells at high electric potentials. Several improvements to the established models of platinum degradation in fuel cell catalysts are proposed, aimed at better description of the diffusion processes involved in particle growth due to Ostwald ripening.

Spatio-temporal coexistence in the cross-diffusion competition system

<p style='text-indent:20px;'>We study a two component cross-diffusion competition system which describes the population dynamics between two biological species. Since the cross-diffusion competition system possesses the so-called population pressure effects, a variety of solution behaviors can be exhibited compared with the classical diffusion competition system. In particular, we discuss on the existence of spatially non-constant time periodic solutions. Applying the center manifold theory and the standard normal form theory, the cross-diffusion competition system is reduced to a two dimensional dynamical system around a doubly degenerate point. As a result, we show the existence of stable time periodic solutions in the system. This means spatio-temporal coexistence between two biological species.

Short-time behavior of continuous-time quantum walks on graphs

Dynamical evolution of systems with sparse Hamiltonians can always be recognized as continuous time quantum walks (CTQWs) on graphs. In this paper, we analyze the short time asymptotics of CTQWs. In recent studies, it was shown that for the classical diffusion process the short time asymptotics of the transition probabilities follows power laws whose exponents are given by the usual combinatorial distances of the nodes. Inspired by this result, we perform a similar analysis for CTQWs both in closed and open systems, including time-dependent couplings. For time-reversal symmetric coherent quantum evolutions, the short time asymptotics of the transition probabilities is completely determined by the topology of the underlying graph analogously to the classical case, but with a doubled power-law exponent. Moreover, this result is robust against the introduction of on-site potential terms. However, we show that time-reversal symmetry breaking terms and non-coherent effects can significantly alter the short time asymptotics. The analytical formulas are checked against numerics, and excellent agreement is found. Furthermore, we discuss in detail the relevance of our results for quantum evolutions on particular network topologies.

Multiple Diffusion-Freezing Mechanisms in Molecular-Hydrogen Films.

Molecular hydrogen is a fascinating candidate for novel bosonic and fermionic superfluids. We have studied diffusion dynamics of thin films of H_{2}, HD, and D_{2} adsorbed on a glass substrate by measurements of elasticity. The elasticity shows multiple anomalies well below the bulk triple point temperature. They are attributed to three different diffusion mechanisms of admolecules and their "freezing" into a localized state: classical thermal diffusion of vacancies, quantum tunneling of vacancies, and diffusion of molecules in the uppermost surface. The surface diffusion is active down to 1K, below which the molecules become localized. This suggests that the surface layer of hydrogen films is on the verge of quantum phase transition to a superfluid state.

Macroscopic Modeling of Water Uptake Behavior of PEDOT:PSS Films.

Poly(3,4-ethylene dioxythiophene)-poly(styrene sulfonate) (PEDOT:PSS) is a widely used conductive polymer (CP) for applications in electronic devices. In the context of transparent electrodes or soft actuators, the uptake of water by PEDOT:PSS is an essential element in the performance of the physical system. We study the water uptake of pure films and films treated with ethylene glycol (EG) commonly used to enhance the electrical properties of PEDOT:PSS. Gravimetric analysis was used to investigate the water sorption–desorption of PEDOT:PSS and its change with the EG treatment for a wide range of configurations (thickness, temperature, and relative humidity). We demonstrate that a simple Fickian model cannot correctly represent the experimental results; we therefore introduce a fully coupled reaction–diffusion scheme. This model describes the transport of diffusing molecules into the polymer film, taking into account, in addition to the classical diffusion mechanism, the reaction between the reactive sites of the polymer network and the water molecules. We demonstrate that solvent treatments have a direct influence on the kinetics of the water uptake of PEDOT:PSS films in terms of diffusivity, solubility, and rate of reaction. The proposed model can be used to accurately predict the water uptake of CP films in the case of complex three-dimensional configurations that are needed for the design of complex actuators.

Simulation of Austenite Formation During Continuous Heating from Low Carbon Martensite with Poly-dispersed Cementite

The nucleation and growth of austenite during continuous heating in plain carbon martensite is simulated using classical nucleation and diffusion growth theories assuming that austenite is nucleated on cementite particles at prior austenite grain boundaries and martensite packet, block, and interlath boundaries. A critical nucleus model on the spherical substrate was modified to take into account the influence of the boundary energy on which cementite particles formed. Simulations were carried out using the particle size distribution of cementite measured in an Fe-0.2 mass pct C alloy heated to near eutectoid temperature (Ae1). Austenite nucleation stopped in a very short time regardless of boundary site or particle size of cementite due to the fast decrease in carbon supersaturation and the depletion of nucleation sites. The fraction of austenite nucleated on cementite at prior austenite boundaries and martensite packet boundaries etc was much greater than that nucleated on cementite at interlath boundaries. While cementite particles dissolved quickly after austenite was nucleated, a large proportion of cementite particles at lath boundaries remained undissolved until they disappeared at 30 °C to 40 °C above Ae1. The evolution of austenite grain size was also simulated after austenitization was completed, and compared with experiment.

Modeling the release of curcumin from microparticles of poly(hydroxybutyrate) [PHB

Polyhydroxybutyrate (PHB) is biodegradable and biocompatible polyester that has been recently used for developing different drug delivery systems (DDS). Microspheres as DDS, consist of a polymeric matrix with a diameter of 1–125 μm which contains a substance entrapped, and then released mainly through diffusion. In order to make DDS viable for commercial applications, it is essential to develop models that describe and predict the substance release kinetic. In this study, microspheres of PHB with curcumin entrapped were synthesized; the release of curcumin was studied and modeled. As a first approach, a physical model was introduced, in which mass transference was considered analogous to heat transference. The proposed model was based on local mass balance through the classical diffusion equation and total mass balance imposed through an integro-differential equation. The total mass balance introduced nonlinearities in the dynamics of the drug within the fluid; therefore, a semi-analytical approach based on the heat balance integral method was applied to solve the proposed model. Finally, by using the experimental rate of release of curcumin, the semi-analytical solutions to the proposed model were compared with the experimental release data, showing the importance of adding mass balance in the nonlinear mathematical model in order to describe more accurately the release kinetics.

Reversible aggregation of particles with short oligomeric sidechains at the surface studied with Langevin dynamics

We investigate nanoscopic aggregation of asphaltene nanoaggregates by means of Brownian dynamics simulations. Nanoaggregates that represent clusters of 6–10 asphaltene molecules are modelled as spherical particles with an attractive core consisting of polyaromatic hydrocarbons rings and short aliphatic tails that protrude from the surface, as in a spherical brush. As two such nanoaggregates approach each other, the steric repulsion that arises from interactions between the tails, is modelled using Alexander-de Gennes theory and Derjaguin approximation, while the attraction upon contact between polyaromatic cores, caused by van der Waals and other interactions is modelled as a Morse potential. The aggregation occurs when the steric barrier is overcome and the cores come into contact. Our simulations show that, with shallow interaction potentials, that is, with weak attractions and low barriers, the nanoaggregates form small agglomerates with ramified structure that are in metastable equilibrium with uniform solutions. These agglomerates can be described with a fractal dimensions below 1.5, which may represent soluble asphaltenes. The increase in attraction strength promotes aggregation and destabilizes the systems as in the case of dilution with a flocculant, for which several aggregation regimes are observed. Increasing the well-depth and/or barrier for nanoaggregate interaction leads to classical diffusion limited and reaction limited regimes. We find that with high barrier the clusters formed are less close-packed, and may represent the weak gel state typical for bitumen gel.

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index γ ∈ [ 0 , 2 ] and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.

Phase-space diffusion of charged particles induced by random fluctuations of relative gyrophases in the presence of finite amplitude circularly polarized electromagnetic waves

A stochastic model to describe the phase-space diffusion of charged particles induced by random fluctuations of relative gyrophases in the presence of parallel propagating, circularly polarized electromagnetic waves is discussed. The perturbation analysis around the equilibrium points of the noiseless system shows the coexistence of classical diffusion and trapping oscillation. Even if the equation of motions for pitch angle does not include noise terms, the pitch angle diffusion occurs due to the noise term in the equation of the relative gyrophase and the existence of the finite amplitude wave. The resultant theory is validated by using numerical results of test particle simulations; when distributions of the relative gyrophases and pitch angle cosines are close to Gaussian, the classical diffusion and trapping oscillation are observed. With increasing wave amplitude and/or the strength of noise, the pitch angle diffusion becomes subdiffusive.

The effect of phonon diffusion on heat transfer

The work is devoted to the development of a tool that allows simulating diffusion in structures where phonons play the main role in the heat transfer (dielectrics, semi-conductors). It is shown that the nature of phonon interaction is fundamentally different from the interactions in classical gases. For the first time, a method for calculating the diffusion of phonons, that takes into account all the important features of their interaction with each other, is presented. A computer simulation of phonon diffusion for a silicon sample in a two-dimensional formulation was carried out. It is shown that the nature of phonon diffusion differs from the classical Brownian diffusion.

Randomized nonlinear software-oriented MDS diffusion layers

Abstract MDS diffusion layers are critical components in the design of symmetric ciphers. In this paper, after introducing some new algebraic structures, we provide new MDS matrices over special types of R-modules. With the help of the proposed methodology, we have more flexibility in designing software-oriented diffusion layers. Most notably, we construct randomized and/or nonlinear MDS diffusion layers, based upon the presented theoretical results, and discuss the resistance of the presented diffusion layers against various kinds of cryptanalysis, compared with classical linear diffusion layers.