Class Of Diffusion Processes Research Articles

Predicting Emergence of Nanoscale Order in Surface Oxides through Preferential Interactivity Parameter.

Diffusion and surface oxidation are critical processes in metal alloy designs and use. Surface oxides provide opportunities to improve material properties or performance beyond bulk alterations. Surface oxidation is, however, often oversimplified into a classical diffusion process. Passivating oxide surfaces are also thought to be lacking in complexity or critical information. A closer look, however, shows inherent complexity with kinetics-driven competition between the elements in the process leading to redox-speciation across a very small (nm) thickness. Questions that remain to be answered for a comprehensive understanding of surface oxides are diverse and call for interdisciplinary approaches. By using the thermodynamics-based Preferential Interactivity Parameter (PIP) alongside kinetic consideration, we show how complexity in these oxides can be predicted allowing us to tailor these thin films. We use our work, and that of others, to illustrate predictability while also highlighting that there is still much more to be done.

Концепция верификации работы клеточных автоматов при вариации геометрических решёток для модели диффузионного процесса

The methodology of cellular automata occupies one of the key positions in the discrete dynamic simulations. The adequacy of the results of solving applied problems is largely determined by the choice of the type of geometric lattice of the cellular automaton. The article presents a formalization and numerical verification of the concept of verification of the operation of a cellular automata model of the classical diffusion process for a set of basic two-dimensional geometric lattices. The approach is based on a direct comparison of the space-time solution of the diffusion equation obtained by the finite element method with the solution of a discrete analogue of this equation based on the operation of a cellular automata algorithm. The software implementation of the cell-automation was performed with the C+ using the Unity platform while the finite element solution of the evolutionary problem of mathematical physics was carried out using the Matlab tools. Cellular automaton lattices were constructed using various cell geometric primitives traditionally used in diffusion modeling applications. Obtained findings indicate that the hexagonal cellular automaton demonstrates the minimum error in the numerical implementation.

Spatiotemporal Observation of Quasi-Ballistic Transport of Electrons in Graphene.

We report spatiotemporal observations of room-temperature quasi-ballistic electron transport in graphene, which is achieved by utilizing a four-layer van der Waals heterostructure to generate free charge carriers. The heterostructure is formed by sandwiching a MoS2 and MoSe2 heterobilayer between two graphene monolayers. Transient absorption measurements reveal that the electrons and holes separated by the type-II interface between MoS2 and MoSe2 can transfer to the two graphene layers, respectively. Transient absorption microscopy measurements, with high spatial and temporal resolution, reveal that while the holes in one graphene layer undergo a classical diffusion process with a large diffusion coefficient of 65 cm2 s-1 and a charge mobility of 5000 cm2 V-1 s-1, the electrons in the other graphene layer exhibit a quasi-ballistic transport feature, with a ballistic transport time of 20 ps and a speed of 22 km s-1, respectively. The different in-plane transport properties confirm that electrons and holes move independently of each other as charge carriers. The optical generation of ballistic charge carriers suggests potential applications for such van der Waals heterostructures as optoelectronic materials.

Rise and Fall, and Slow Rise Again, of Operator Entanglement under Dephasing.

The operator space entanglement entropy, or simply "operator entanglement" (OE), is an indicator of the complexity of quantum operators and of their approximability by matrix product operators (MPOs). We study the OE of the density matrix of 1D many-body models undergoing dissipative evolution. It is expected that, after an initial linear growth reminiscent of unitary quench dynamics, the OE should be suppressed by dissipative processes as the system evolves to a simple stationary state. Surprisingly, we find that this scenario breaks down for one of the most fundamental dissipative mechanisms: dephasing. Under dephasing, after the initial "rise and fall," the OE can rise again, increasing logarithmically at long times. Using a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, we demonstrate that this growth is inherent to a U(1) conservation law. We argue that in an XXZ spin model and a Bose-Hubbard model the OE grows universally as 1/4log_{2}t at long times and as 1/2log_{2}t for a Fermi-Hubbard model. We trace this behavior back to anomalous classical diffusion processes.

Unbiased estimation using a class of diffusion processes

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (Rd,B(Rd)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process {Xt}t∈[0,1] on Rd, d∈N0. This latter process is constructed such that, starting at X0=0, one has X1∼π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, [10,16] consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t. π. We show that for this methodology to achieve a mean square error of O(ϵ2), for arbitrary ϵ>0, the associated cost is O(ϵ−5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t. π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ϵ2), the associated cost (which is random) is, with high probability, O(ϵ−2|log⁡(ϵ)|2+δ), for any δ>0. We implement our method on several examples including Bayesian inverse problems.

Simultaneous determination of different class of parameters for a diffusion equation from a single measurement

This article is devoted to the simultaneous resolution of several inverse problems, among the most important formulation of inverse problems for partial differential equations, stated for some class of diffusion equations from a single boundary measurement. Namely, we consider the simultaneous unique determination of several class of coefficients, some internal sources (a source term and an initial condition) and the fractional order in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium) and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection–diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation.

Unbiased Estimation Using a Class of Diffusion Processes

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) $\pi$ a given, general probability measure on $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schr\"odinger-F\"ollmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process $\{X_t\}_{t\in [0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process is constructed such that, starting at $X_0=0$, one has $X_1\sim \pi$. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, \cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t.~$\pi$. We show that for this methodology to achieve a mean square error of $\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$, the associated cost is $\mathcal{O}(\epsilon^{-5})$. We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t.~$\pi$, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of $\mathcal{O}(\epsilon^2)$, the associated cost is, with high probability, $\mathcal{O}(\epsilon^{-2}|\log(\epsilon)|^{2+\delta})$, for any $\delta>0$. We implement our method on several examples including Bayesian inverse problems.

Layer-by-layer anionic diffusion in two-dimensional halide perovskite vertical heterostructures.

Anionic diffusion in a soft crystal lattice of hybrid halide perovskites affects their stability, optoelectronic properties and the resulting device performance. The use of two-dimensional (2D) halide perovskites improves the chemical stability of perovskites and suppresses the intrinsic anionic diffusion in solid-state devices. Based on this strategy, devices with an enhanced stability and reduced hysteresis have been achieved. However, a fundamental understanding of the role of organic cations in inhibiting anionic diffusion across the perovskite-ligand interface is missing. Here we demonstrate the first quantitative investigation of the anionic interdiffusion across atomically flat 2D vertical heterojunctions. Interestingly, the halide diffusion does not follow the classical diffusion process. Instead, a 'quantized' layer-by-layer diffusion model is proposed to describe the behaviour of the anionic migration in 2D halide perovskites. Our results provide important insights into the mechanism of anionic diffusion in 2D perovskites and provide a new materials platform with an enhanced stability for heterostructure integration.

The rationale for the method of thermochemical processing of beet chips before extraction of sucrose

The objective of this thesis is to ensure utmost performance of the diffusion process by choosing the most effective components of chemical heat treatment of beet chips. Aqueous solutions of ammonia sulphates (NH4)2SO4, aluminium sulphate Al2(SO4)3, calcium sulphate CaSO4 have been considered as reagents to process chips. The classical diffusion process method has been used as a benchmarking option. It has been deduced from experiments that beet tissue treatment with solutions of the proposed reagents heated up to 73 ° C enables a high degree of cell plasmotosis, enhances its permeability and intensifies sucrose transfer to the extractant during heavy convectional leaching. The process is strongly enhanced if chips are treated with reagent solutions following electrochemical activation, which is confirmed by the study of the beet tissue fine structure after its various chemical heat treatments. Rational conditions of solution electrochemical activation for beet chips chemical heat treatment prior to extraction have been selected experimentally. 3 patents of the Russian Federation for the diffusion juice process have been obtained based on the results of the studies conducted.

On simulating the short and long memory of ergodic Markov and Non-Markov genetic diffusion processes on the long run

Abstract The theory of classical ergodic Markov genetic diffusion process, Feller genetic diffusion, and the genetic drift diffusion with and without selection and mutation for a single unlinked locus with two alleles, Kimura models, is studied. These genetic problems are numerically investigated on the long run. The approximates solutions of these models are described as the conditional probabilities to find the specific gene with frequencies between zero and one at generation T . As a surprise, we notice that the summation of these approximate solutions lose their unity rapidly as the number of generations increase than five, i.e. T > > 5 . The earlier biologists did due the reason to the migration, immigration and death of individuals. Till now, there is no mathematical proof to this phenomenon. In this paper, we solve this problem by extending the first order time derivative to the time fractional derivative, to study the effect of the memory on these models. This extension ensures that the summations of the approximate solutions, i.e. of the Non-Markov cases, are one at any number of generations. The time evolution of the approximate solutions are simulated and compared with other published papers, for different values of generations and for the Markov and Non-Markov cases with different values of the fractional order β . The convergence of the approximate descrete solutions and the reversibility property of these stochastic processes for both the Markov cases and Non-Markov cases are also numerically simulated and discussed.

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.

Turning Drops into Bubbles: Cavitation by Vapor Diffusion through Elastic Networks.

Some members of the vegetal kingdom can achieve surprisingly fast movements making use of a clever combination of evaporation, elasticity, and cavitation. In this process, enthalpic energy is transformed into elastic energy and suddenly released in a cavitation event which produces kinetic energy. Here, we study this unusual energy transformation by a model system: A droplet in an elastic medium shrinks slowly by diffusion and eventually transforms into a bubble by a rapid cavitation event. The experiments reveal the cavity dynamics over the extremely disparate timescales of the process, spanning 9 orders of magnitude. We model the initial shrinkage as a classical diffusive process, while the sudden bubble growth and oscillations are described using an inertial-(visco)elastic model, in excellent agreement with the experiments. Such a model system could serve as a new paradigm for motile synthetic materials.

Yule-Walker type estimator of first-order time-varying periodic bilinear differential model for stochastic processes

This paper, studies the class of diffusion processes generated by a first-order continuous-time bilinear processes (COBL(1, 1)) with time-varying coefficients. So, we used the It formula approach for examining the structure of the process and its powers. In time-invariant case, an expression of the moments of any order are given and the continuous autoregressive representation of such version is given, in particular the moments properties of some specifications are however derived. Based on these results we are able to examine the statistical properties as well as we develop an estimation method of the process via the so-called Yule-Walker (YW) type algorithm which relates with unknown parameters of CAR representation. The method is illustrated by a Monte Carlo study and applied to modeling the electricity consumption sampled at each 15 mn in Algeria.

Fractional Diffusion Models for the Atmosphere of Mars

The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative.

Real-time and encryption efficiency improvements of simultaneous fusion, compression and encryption method based on chaotic generators

In this paper, enhancement of an existing optical simultaneous fusion, compression and encryption (SFCE) scheme in terms of real-time requirements, bandwidth occupation and encryption robustness is proposed. We have used and approximate form of the DCT to decrease the computational resources. Then, a novel chaos-based encryption algorithm is introduced in order to achieve the confusion and diffusion effects. In the confusion phase, Henon map is used for row and column permutations, where the initial condition is related to the original image. Furthermore, the Skew Tent map is employed to generate another random matrix in order to carry out pixel scrambling. Finally, an adaptation of a classical diffusion process scheme is employed to strengthen security of the cryptosystem against statistical, differential, and chosen plaintext attacks. Analyses of key space, histogram, adjacent pixel correlation, sensitivity, and encryption speed of the encryption scheme are provided, and favorably compared to those of the existing crypto-compression system. The proposed method has been found to be digital/optical implementation-friendly which facilitates the integration of the crypto-compression system on a very broad range of scenarios.

Maximal Exponential Inequalities for Certain Diffusion Processes

In this paper, we prove maximal exponential inequalities for a class of diffusion processes under certain conditions. The present results extend the well-known maximal inequalities of the power-type proved in the case of a Brownian motion, and in the geometric Brownian motion case. The method of proof is essentially based on explicit forms of optimal stopping problems related to those treated by Shepp and Shiryaev [Ann. Probab. Appl., 3 (1993), pp. 631--640], but considered here with a multiplicative reward functional. The key tool in the proof of one of our main results is a comparison principle for solutions of a system of first-order nonlinear differential equations. This is one of the novel features in this paper.

Anomalous Diffusion with an Irreversible Linear Reaction and Sorption-Desorption Process

We investigate the diffusion of two different species in a semi-infinite medium considering the presence of linear reaction terms. The dynamics for these species is governed by fractional diffusion equations. We also consider the presence of an adsorption-desorption boundary condition. The solutions for this system are found in terms of the H function of Fox and by analyzing the behavior of the mean square displacement a rich class of diffusion processes is verified. In this sense, we show how the surface effects modify the bulk dynamics and promote an anomalous diffusion of system.

Optimization of Vacuum Impregnation with Calcium Lactate of Minimally Processed Melon and Shelf-Life Study in Real Storage Conditions.

Vacuum impregnation (VI) is a processing operation that permits the impregnation of fruit and vegetable porous tissues with a fast and more homogeneous penetration of active compounds compared to the classical diffusion processes. The objective of this research was to investigate the impact on VI treatment with the addition of calcium lactate on qualitative parameters of minimally processed melon during storage. For this aim, this work was divided in 2 parts. Initially, the optimization of process parameters was carried out in order to choose the optimal VI conditions for improving texture characteristics of minimally processed melon that were then used to impregnate melons for a shelf-life study in real storage conditions. On the basis of a 23 factorial design, the effect of Calcium lactate (CaLac) concentration between 0% and 5% and of minimum pressure (P) between 20 and 60 MPa were evaluated on color and texture. Processing parameters corresponding to 5% CaLac concentration and 60 MPa of minimum pressure were chosen for the storage study, during which the modifications of main qualitative parameters were evaluated. Despite of the high variability of the raw material, results showed that VI allowed a better maintenance of texture during storage. Nevertheless, other quality traits were negatively affected by the application of vacuum. Impregnated products showed a darker and more translucent appearance on the account of the alteration of the structural properties. Moreover microbial shelf-life was reduced to 4 d compared to the 7 obtained for control and dipped samples.

Identification of unstable fixed points for randomly perturbed dynamical systems with multistability

Multistability, especially bistability, is one of the most important nonlinear phenomena in deterministic and stochastic dynamics. The identification of unstable fixed points for randomly perturbed dynamical systems with multistability has drawn increasing attention in recent years. In this paper, we provide a rigorous mathematical theory of the previously proposed data-driven method to identify the unstable fixed points of multistable systems. Specifically, we define a family of statistics which can be estimated by practical time-series data and prove that the local maxima of this family of statistics will converge to the unstable fixed points asymptotically. During the proof of the above result, we obtain two mathematical by-products which are interesting in their own right. We prove that the downhill timescale for randomly perturbed dynamical systems is log⁡(1/ϵ), different from the uphill timescale of eV/ϵ for some V>0 predicted by the Freidlin–Wentzell theory. Moreover, we also obtain an Lp maximum inequality for randomly perturbed dynamical systems and a class of diffusion processes.

Alternative to beta coefficients in the context of diffusions

We develop an alternative to the beta coefficient of the CAPM theory. We show the link between this notion and the Wiener chaos expansion of the underlying processes. In the setting of Markov diffusions, we define the drift-neutral beta, which is the quantity of benchmark such that the resulting portfolio is immune to an infinitesimal change of drift on the Brownian motion driving the benchmark. Our approach yields a coefficient which in many practical cases depends on the initial values of both the portfolio and its benchmark. It can also be used to take into account extreme risks and not only the variance. We study several classical diffusion processes and give a full analysis in the case of Jacobi processes. Examples with credit indices show the efficiency of the method in hedging a portfolio.