Continuous Random Walk Research Articles

Particle deposition modeling in the secondary side of a steam generator bundle model

A steam generator (SG) tube rupture (SGTR) model is studied in this paper. This model based on a experimental facility called Aerosol Trapping In a Steam Generator (ARTIST), which is a model of a scaled steam generator tube bundle consisting of 270 tubes and a guillotine tube to address aerosol deposition phenomena on two different scales: near the tube break, where the gas velocities and turbulence are very intensive, and far away from the break, where the flow velocities are three orders of magnitude lower.Owing to complexity of the flow, 3D simulations with highly resolved computational mesh near the break were done. First, the flow inside an isolated tube with a guillotine tube break has been studied in the framework of Reynolds Averaged Navier Stokes (RANS) approach. The next part is devoted to the simulation of an inclined gas jet entering the SG tube bundle via the guillotine tube breach with more advanced CFD tools. In particular, Detached Eddy Simulation (DES) and RANS are applied to tackle the wide range of flow scales. Flow field velocity comparison showed that DES results are reproducing wavy structure of the flow field in far field from the break observed in experiment.Particle transport and deposition is modelled by Lagrangian continuous random walk (CRW) model, which has been developed and validated previously. It is found that the DES combined with the CRW to supply fluctuating velocity components predicts deposition rates that are generally within the scatter of the measured data. Monodisperse, spherical SiO2 particles with AMMD = 1.4μm were used as aerosol particles in simulations.To be economically feasible, the computations were made with the open source CFD code OpenFOAM. Comparison of the calculated flow with the experimental axial velocity distribution data at different vertical levels has been performed.

RANS modeling for particle transport and deposition in turbulent duct flows: Near wall model uncertainties

Dust accumulation in the primary system of a (V)HTR is identified as one of the foremost concerns during a potential accident. Several numerical efforts have focused on the use of RANS methodology to better understand the complex phenomena of fluid–particle interaction at various flow conditions. In the present work, several uncertainties relating to the near-wall modeling of particle transport and deposition are addressed for the RANS approach. The validation analyses are performed in a fully developed turbulent duct flow setup. A standard k−ɛ turbulence model with enhanced wall treatment is used for modeling the turbulence. For the Lagrangian phase, the performance of a continuous random walk (CRW) model and a discrete random walk (DRW) model for the particle transport and deposition are assessed. For wall bounded flows, it is generally seen that accounting for near wall anisotropy is important to accurately predict particle deposition. The various near-wall correlations available in the literature are either derived from the DNS data or from the experimental data. A thorough investigation into various near-wall correlations and their applicability for accurate particle deposition predictions are assessed. The main outcome of the present work is a well validated turbulence model with optimal near-wall modeling which provides realistic particle deposition predictions.

A simple stochastic quadrant model for the transport and deposition of particles in turbulent boundary layers

We present a simple stochastic quadrant model for calculating the transport and deposition of heavy particles in a fully developed turbulent boundary layer based on the statistics of wall-normal fluid velocity fluctuations obtained from a fully developed channel flow. Individual particles are tracked through the boundary layer via their interactions with a succession of random eddies found in each of the quadrants of the fluid Reynolds shear stress domain in a homogeneous Markov chain process. In this way, we are able to account directly for the influence of ejection and sweeping events as others have done but without resorting to the use of adjustable parameters. Deposition rate predictions for a wide range of heavy particles predicted by the model compare well with benchmark experimental measurements. In addition, deposition rates are compared with those obtained from continuous random walk models and Langevin equation based ejection and sweep models which noticeably give significantly lower deposition rates. Various statistics related to the particle near wall behavior are also presented. Finally, we consider the model limitations in using the model to calculate deposition in more complex flows where the near wall turbulence may be significantly different.

Large Deviations for Stationary Probabilities of a Family of Continuous Time Markov Chains via Aubry–Mather Theory

In the present paper, we consider a family of continuous time symmetric random walks indexed by \(k\in \mathbb {N}\), \(\{X_k(t),\,t\ge 0\}\). For each \(k\in \mathbb {N}\) the matching random walk take values in the finite set of states \(\Gamma _k=\frac{1}{k}(\mathbb {Z}/k\mathbb {Z})\); notice that \(\Gamma _k\) is a subset of \(\mathbb {S}^1\), where \(\mathbb {S}^1\) is the unitary circle. The infinitesimal generator of such chain is denoted by \(L_k\). The stationary probability for such process converges to the uniform distribution on the circle, when \(k\rightarrow \infty \). Here we want to study other natural measures, obtained via a limit on \(k\rightarrow \infty \), that are concentrated on some points of \(\mathbb {S}^1\). We will disturb this process by a potential and study for each \(k\) the perturbed stationary measures of this new process when \(k\rightarrow \infty \). We disturb the system considering a fixed \(C^2\) potential \(V: \mathbb {S}^1 \rightarrow \mathbb {R}\) and we will denote by \(V_k\) the restriction of \(V\) to \(\Gamma _k\). Then, we define a non-stochastic semigroup generated by the matrix \(k\,\, L_k + k\,\, V_k\), where \(k\,\, L_k \) is the infinifesimal generator of \(\{X_k(t),\,t\ge 0\}\). From the continuous time Perron’s Theorem one can normalized such semigroup, and, then we get another stochastic semigroup which generates a continuous time Markov Chain taking values on \(\Gamma _k\). This new chain is called the continuous time Gibbs state associated to the potential \(k\,V_k\), see (Lopes et al. in J Stat Phys 152:894–933, 2013). The stationary probability vector for such Markov Chain is denoted by \(\pi _{k,V}\). We assume that the maximum of \(V\) is attained in a unique point \(x_0\) of \(\mathbb {S}^1\), and from this will follow that \(\pi _{k,V}\rightarrow \delta _{x_0}\). Thus, here, our main goal is to analyze the large deviation principle for the family \(\pi _{k,V}\), when \(k \rightarrow \infty \). The deviation function \(I^V\), which is defined on \( \mathbb {S}^1\), will be obtained from a procedure based on fixed points of the Lax–Oleinik operator and Aubry–Mather theory. In order to obtain the associated Lax–Oleinik operator we use the Varadhan’s Lemma for the process \(\{X_k(t),\,t\ge 0\}\). For a careful analysis of the problem we present full details of the proof of the Large Deviation Principle, in the Skorohod space, for such family of Markov Chains, when \(k\rightarrow \infty \). Finally, we compute the entropy of the invariant probabilities on the Skorohod space associated to the Markov Chains we analyze.

Anomalies in non-Markovian quantum dynamics

The non-Markovian dynamics of general finite dimensional open quantum systems is studied. In the framework of the continuous time quantum random walk inverse power law tailed distributions of the waiting times between two consecutive interactions with the external environment are considered. Hindered time evolution appears over long time scales if the distribution of the long waiting times increases up to the critical decay . More generally, arbitrarily slow relaxations to the asymptotic configurations are found in the scenario of dynamical maps of a qudit. Similar features characterize the time evolution of the entanglement of X states.

Universality of trap models in the ergodic time scale

Consider a sequence of possibly random graphs $G_{N}=(V_{N},E_{N})$, $N\ge1$, whose vertices’s have i.i.d. weights $\{W^{N}_{x}:x\in V_{N}\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^{N}_{t}$, $t\ge0$, be a continuous time simple random walk on $G_{N}$ which waits a mean $W^{N}_{x}$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge2$, random $d$-regular graphs and the largest component of super-critical Erdős–Renyi random graphs.

Coherent random walks in free space

Two-dimensional continuous time quantum random walks (CTQRW) are physical processes where quantum particles simultaneously evolve in different permissible directions within discrete graphs. In order to force the quantum walkers (QWs) to evolve in such a fashion, one generally requires periodic potentials. Here, we demonstrate that two-dimensional CTQRW can be generated in free space by properly tailoring the initial wave functions. We analytically show that within a certain spatial region the arising probability distribution quantitatively resembles the probability pattern exhibited by a QW traversing a periodic lattice potential. These theoretical predictions were experimentally verified using classical laser light, appropriately shaped by a spatial light modulator. Expanding the presented results to the case of multiple walkers may open new possibilities in quantum information technology using bulk optics.

Itô formula for one-dimensional continuous-time quantum random walk

In the paper, we first extend the discrete time quantum random walk (DQRW) to continuous time quantum random walk (CQRW). Then we establish an Itô formula for the one-dimensional CQRW. As an application, we present Tanaka’s formula with the Itô formula and characterize the relation between quantum local time Lta and Dirac delta function δ. Finally, we discuss quantum stochastic calculus for the CQRW.

Continuous Time Open Quantum Random Walks and Non-Markovian Lindblad Master Equations

A new type of quantum random walks, called Open Quantum Random Walks, has been developed and studied in Attal et al. (Open quantum random walks, preprint) and (Central limit theorems for open quantum random walks, preprint). In this article we present a natural continuous time extension of these Open Quantum Random Walks. This continuous time version is obtained by taking a continuous time limit of the discrete time Open Quantum Random Walks. This approximation procedure is based on some adaptation of Repeated Quantum Interactions Theory (Attal and Pautrat in Annales Henri Poincare Physique Theorique 7:59–104, 2006) coupled with the use of correlated projectors (Breuer in Phys Rev A 75:022103, 2007). The limit evolutions obtained this way give rise to a particular type of quantum master equations. These equations appeared originally in the non-Markovian generalization of the Lindblad theory (Breuer in Phys Rev A 75:022103, 2007). We also investigate the continuous time limits of the quantum trajectories associated with Open Quantum Random Walks. We show that the limit evolutions in this context are described by jump stochastic differential equations. Finally we present a physical example which can be described in terms of Open Quantum Random Walks and their associated continuous time limits.

Kinetic study of the heterogeneous photocatalysis of porous nanocrystalline TiO₂ assemblies using a continuous random walk simulation.

The continuous time random walk (CTRW) simulation was used to study the photocatalytic kinetics of nanocrystalline (nc)-TiO2 assemblies in this research. nc-TiO2 assemblies, such as nc-TiO2 porous films and nc-TiO2 hierarchical structures, are now widely used in photocatalysis. The nc-TiO2 assemblies have quasi-disordered networks consisting of many tiny nanoparticles, so the charge transport within them can be studied by CTRW simulation. We considered the experimental facts that the holes can be quickly trapped and transferred to organic species just after photogeneration, and the electrons transfer to O2 slowly and accumulate in the conduction band of TiO2, which is believed to be the rate-limiting process of the photocatalysis under low light intensity and low organic concentration. Due to the existence of numerous traps, the electron transport within the nc-TiO2 assemblies follows a multi-trapping (MT) mechanism, which significantly limits the electron diffusion speed. The electrons need to undergo several steps of MT transport before transferring to oxygen, so it is highly important that the electron transport in nc-TiO2 networks is determined for standard photocatalytic reactions. Based on the MT transport model, the transient decays of photocurrents during the photocatalytic oxidation of formic acid were studied by CTRW simulation, and are in good accordance with experiments. The steady state photocatalysis was also simulated. The effects of organic concentration, light intensity, temperature, and nc-TiO2 crystallinity on the photocatalytic kinetics were investigated, and were also consistent with the experimental results. Due to the agreement between the simulation and the experiments for both the transient and the steady state photocatalysis, the MT charge transport should be an important mechanism that controls the kinetics of recombination and photocatalysis in nc-TiO2 assemblies. Also, our research provides a new methodology to study the photocatalytic dynamics from the random event viewpoint, which can be revised to investigate the kinetic processes of other kinds of material.

Classification of fractional order biomarkers for anomalous diffusion using q-space entropy.

In this study, we applied continuous random walk theory (CTRW) to develop a new model that characterizes anomalous diffusion in magnetic resonance imaging experiments. Furthermore, we applied a classification scheme based on information theoretic a techniques to characterize the degree of heterogeneity and complexity in biological tissues. From a CTRW approach, the Fourier transform of the generalized solution to the diffusion equation comes in the form of the Mittag-Leffler function. In this solution form, the relative stochastic uncertainty in the diffusion process can be computed with spectral entropy. We interrogated both white and gray matter regions of a fixed rat brain with diffusion - weighted magnetic resonance imaging experiments up to 26,000 s/mm² by independently weighting q and Δ. to investigate the effects on the diffusion phenomena. Our model fractional order parameters, α and β, and entropy measure, H(q, Δ), differentiated between tissue types and extracted differing information within a region of interest based on the type of diffusion experiment performed. By combining fractional order modeling and information theory, new and powerful biomarkers are available to characterize tissue microstructure and provide contextual information about the anatomical complexity.

Numerical analysis of stochastic dispersion of micro-particles in turbulent flows in a realistic model of human nasal/upper airway

Turbulent flow field and particle deposition in a realistic model of human upper airway system including: nasal cavity, nasopharynx, oropharynx, larynx and trachea were analyzed numerically. The Lagrangian approach was used to find the trajectories of micron-size particles for the breathing rates of 30, 45 and 60l/min. The continuous phase flow was evaluated using a RANS (Reynolds Average Navier-Stokes) turbulence model and the effect of turbulent fluctuations on particle trajectories was modeled using a continuous random walk (CRW) stochastic model, which is based on the normalized Langevin equation. The total and regional deposition fractions of micro-particles were evaluated and the effect of turbulent fluctuations on the particle deposition rate was investigated.

Continuous random walks and fractional powers of operators

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes. This short paper represents a bridge between probabilists and researchers working in PDEʼs.

Numerical simulation of multilayer deposition in an obstructed channel flow

Simulation of multilayer deposition of dry aerosol particles in turbulent flows has gained a growing interest in various industrial and research applications. The multilayer deposition of carbonaceous aerosol particles in a turbulent channel flow obstructed by a succession of square ribs is here numerically investigated. The multilayer particle bed growth on the various wall surfaces affects the air flow, which in turn affects the overall deposition rate. An iterative numerical procedure is therefore suggested to simulate the evolution of the graphite layer. The iterative process used to reproduce the layer build-up is decomposed as follows: Reynolds-Avergared Navier Stokes is employed to generate the flow field. The turbulent dispersion of the particles is reproduced through the use of a continuous random walk model. After statistically sufficient deposition of particulate matter, the layer build-up is computed using mechanics of dry granular material. The layer build-up model shows good agreement with data obtained from experimental tests carried out on-site.

Fokker-Planck equation with memory: the cross over from ballistic to diffusive processes in many particle systems and incompressible media

The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non-Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times, these solutions agree with the predictions of the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial.

Probabilistic Extention of the Cumulative Prospect Theory

A number of experiments indicate probabilistic preferences in cases where no one alternative is absolutely optimal. The task of predicting the choice of one of the alternatives among multiple alternatives is then practically important and not trivial. It can occur in situations of choice under risk when no one lottery stochastically dominates others.For risky lotteries there are several complicated models of probabilistic binary preference. For the first time, we herein propose the probabilistic extension of the cumulative prospect theory (CPT). The presented visual graphic justification of this model is intuitively clear and does not use sophisticated cumulative summing or a Choquet integral. Here we propose a model of selecting from a set of alternatives by continuous Markov random walks. It makes predicting the results of a choice easy because it fully uses dates received by probabilistic extension of СPT. The proposed methods are quite simple and do not require a large amount of data for practical use.

Non-Fickian dispersion of groundwater age.

We expand the governing equation of groundwater age to account for non-Fickian dispersive fluxes using continuous random walks. Groundwater age is included as an additional (fifth) dimension on which the volumetric mass density of water is distributed and we follow the classical random walk derivation now in five dimensions. The general solution of the random walk recovers the previous conventional model of age when the low order moments of the transition density functions remain finite at their limits and describes non-Fickian age distributions when the transition densities diverge. Previously published transition densities are then used to show how the added dimension in age affects the governing differential equations. Depending on which transition densities diverge, the resulting models may be nonlocal in time, space, or age and can describe asymptotic or pre-asymptotic dispersion. A joint distribution function of time and age transitions is developed as a conditional probability and a natural result of this is that time and age must always have identical transition rate functions. This implies that a transition density defined for age can substitute for a density in time and this has implications for transport model parameter estimation. We present examples of simulated age distributions from a geologically based, heterogeneous domain that exhibit non-Fickian behavior and show that the non-Fickian model provides better descriptions of the distributions than the Fickian model.

Open System Approach to the Internal Dynamics of a Model Multilevel Molecule

A model multilevel molecule described by two sets of rotational internal energy levels of different parity and degenerate ground states, coupled by a constant interaction, is considered, by assuming that the random collisions in a gas of identical molecules, provoke transitions between adjacent energy levels of the same parity. The prescriptions of the continuous time quantum random walk is applied to the single molecule, interpreted as an open quantum system, and the master equation driving its internal dynamics is built for a general distribution of random collision times. Over estimated long time scales, the dynamics of the coherence terms and the populations of the energy levels is evaluated analytically for relevant classes of non-Poissonian distributions of the collision times. Inverse power law relaxations and the inverse Zeno effect emerge over estimated long time scales and the stable asymptotic equilibrium configuration results to be independent of the distribution of the collision times.

A Lagrangian particle random walk model for hybrid RANS/LES turbulent flows

A continuous random walk (CRW) turbulent diffusion model was adapted for Lagrangian particles within gas flowfields simulated by hybrid RANS/LES methodologies. The methodology was designed to model all the particle diffusion in RANS regions and model only the sub-grid diffusion in LES regions. In the RANS approach, the mean flowfield and the turbulent time- and length-scales are obtained with a k–ω (Menter SST) turbulence model. These values are used with a discrete stochastic equation to compute instantaneous gas velocity along an individual particle trajectory. Experimental results for turbulent diffusion of particles in a homogeneous wake flow were first used to calibrate the RANS model. The stochastic diffusion model was then extended to utilize the Nichols–Nelson k–ω hybrid RANS/LES turbulence model in the unsteady three-dimensional wake of a cylinder. In particular, the flow at a Mach number of 0.1 and Reynolds number (ReD) of 800 was computed with a 5th-order upwind-biased scheme. The discrete stochastic equation was used to compute sub-grid fluctuations, which could be added to the resolved velocity field, and specifically took into account combined effects of particle inertia and non-homogeneous turbulence. The combination of resolved diffusion and sub-grid diffusion compared quite reasonably with diffusion based on Direct Numerical Simulation of the Navier–Stokes equations. The results indicate that eddy-crossing effects and inertia-based drift corrections can be critical, even when most of the kinetic energy is captured with the resolved-scales of an LES approach.

The sequential alternative search as a continuous Markov random walk

The course of the process is analyzed for the choice by an individual of alternatives from a few alternatives, among which he has paired preferences, possibly intransitive and probabilistic. It is supposed that the individual carries out the choice with the aid of the process of the sequential exhaustive search of alternatives, on which a number of conditions (axioms) are imposed. It is shown that in this case the process is modeled by the continuous Markov random walk in the set of alternatives; the intensities of transitions are equal to the comparative utility function; the choice result will be probabilistic and will correspond to the established probability distribution in the random walk. It is shown that the choice result will be the Nash equilibrium in the game, which is set up on the paired preferences of the individual.