Stochastic Particle Model Research Articles

Algorithms for Propagating Uncertainty Across Heterogeneous Domains

We address an important research area in stochastic multiscale modeling, namely, the propagation of uncertainty across heterogeneous domains characterized by partially correlated processes with vastly different correlation lengths. This class of problems arises very often when computing stochastic PDEs and particle models with stochastic/stochastic domain interaction but also with stochastic/deterministic coupling. The domains may be fully embedded, adjacent, or partially overlapping. The fundamental open question we address is the construction of proper transmission boundary conditions that preserve global statistical properties of the solution across different subdomains. Often, the codes that model different parts of the domains are black box and hence a domain decomposition technique is required. No rigorous theory or even effective empirical algorithms have yet been developed for this purpose, although interfaces defined in terms of functionals of random fields (e.g., multipoint cumulants) can overco...

Classical and Quantum Mechanical Models of Many-Particle Systems

This meeting was focused on recent results on the mathematical analysis of many-particle systems, both classical and quantum-mechanical in scaling regimes such that the methods of kinetic theory can be expected to apply. Thus, the Boltzmann equation is in many ways the central equation investigated in much of the research presented and discussed at this meeting, but the range of topics naturally extended from this center to include other non-linear partial differential and integro-differential equations, especially macroscopic/fluid-dynamical limits of kinetic equations modeling the dynamics of many-particle systems. A significant subset of the talks focused on propagation of chaos, and the validation and derivation of kinetic equations from underlying stochastic particle models in which there has been much progress and activity. Models were discussed with applications not only in physics, but also engineering, and mathematical biology. While there were a number of new participants, especially younger researchers, an interesting aspect of the conference was the number of talks presenting progress that had its origins in the previous meeting in this series held in 2010.

Correction: Collective Cell Motion in an Epithelial Sheet Can be Quantitatively Described by a Stochastic Interacting Particle Model

Correction: Collective Cell Motion in an Epithelial Sheet Can be Quantitatively Described by a Stochastic Interacting Particle Model

Modeling selective local interactions with memory: Motion on a 2D lattice

We consider a system of particles that simultaneously move on a two-dimensional periodic lattice at discrete times steps. Particles remember their last direction of movement and may either choose to continue moving in this direction, remain stationary, or move toward one of their neighbors. The form of motion is chosen based on predetermined stationary probabilities. Simulations of this model reveal a connection between these probabilities and the emerging patterns and size of aggregates. In addition, we develop a reaction–diffusion master equation from which we derive a system of ODEs describing the dynamics of the particles on the lattice. Simulations demonstrate that solutions of the ODEs may replicate the aggregation patterns produced by the stochastic particle model. We investigate conditions on the parameters that influence the locations at which particles prefer to aggregate. This work is a two-dimensional generalization of Galante and Levy (2012), in which the corresponding one-dimensional problem was studied.

Active fluctuation symmetries

In contrast with the understanding of fluctuation symmetries for entropy production, similar ideas applied to the time-symmetric fluctuation sector have been less explored. Here we give detailed derivations of time-symmetric fluctuation symmetries in boundary-driven particle systems such as the open Kawasaki lattice gas and the zero-range model. As a measure of time-symmetric dynamical activity over time T we count the difference (Nℓ − Nr)/T between the number of particle jumps in or out at the left edge and those at the right edge of the system. We show that this quantity satisfies a fluctuation symmetry from which we derive a new Green–Kubo-type relation. It will follow then that the system is more active at the edge connected to the particle reservoir with the largest chemical potential. We also apply these exact relations derived for stochastic particle models to a deterministic case, the spinning Lorentz gas, where the symmetry relation for the activity is checked numerically.

On the integrability of zero-range chipping models with factorized steady states

The conditions of the integrability of general zero range chipping models with factorized steady states, which were proposed in Evans et al (2004 J. Phys. A: Math. Gen. 37 L275), are examined. We find a three-parametric family of hopping probabilities for the models solvable by the Bethe ansatz, which includes most of known integrable stochastic particle models as limiting cases. The solution is based on the quantum binomial formula for two elements of an associative algebra obeying generic homogeneous quadratic relations, which is proved as a byproduct. We use the Bethe ansatz to solve an eigenproblem for the transition matrix of the Markov process. On its basis, we conjecture an integral formula for the Green function of the evolution operator for the model on an infinite lattice and derive the Bethe equations for the spectrum of the model on a ring.

A Stochastic Single-Particle Lagrangian Model for the Concentration Fluctuations in a Plume Dispersing Inside an Urban Canopy

A new approach for estimating concentration fluctuations intensity in dense built-up environments using a Lagrangian stochastic (LS) particle model is described. Following past success in modelling the dynamics of concentration variance as a diffusion-advection process, the ensemble-averaged concentration variance is represented by particles that advect and diffuse throughout the computational domain. The calculation of the concentration variance is addressed by assuming an appropriate distribution of effective variance sources for a given mean concentration field. Dissipation is treated by allowing the variance carried by every particle to decay exponentially with a locally-estimated decay time. The approach has the benefit of easily handling complex boundary conditions. It can also be easily and naturally implemented as an extension to an existing LS model, which is used for mean concentration estimations. The method differs from existing two-particle methods that demand knowledge of the structure function of the flow. It is also more computationally efficient than micro-mixing approaches that involve maintaining high population levels of particles in every grid volume. The model is compared with high frequency concentration measurements, taken as part of the JU2003 (Joint Urban 2003) experiment that was carried out in Oklahoma City. Good agreement is observed.

Dynamics of the Eyeblink EMG at Global and Microscopic Scales

Eyeblink electromyogram (EMG) is a noninvasive and reliable tool for evaluating information processing at different levels of the central nervous system.The recently developed method of fragmentary decompositioncreates the model of a single trial eyeblink EMG as the series of consecutive, partly overlapping components, with the generic mass potential being the basis element. Here we express this model in terms of underlying cellular processes. To our knowledge, this is the first study where the dynamics of a mass potential at global scale is deduced from a stochastic particle model of the ion movements at microscopic scale. We consider generation of the eyeblink EMG as a stimulus- induced creation of an extracellular dipole by the movements of ions at the microscopic scale. These processes are described in terms of an equivalent circuit which is composed of novel circuit elements named sourceoid, sinkoid, and dipoloid. The corresponding equations are formulated in terms of nonhomogeneous birth-and-death processes. The theory brings together the deterministic and stochastic factors underlying the genesis of eyeblink EMG with the minimum number of free parameters. No matter how complex are the particle systems, just a few global scale parameters accumulate essential aspects of microscale activities which appear as significant variables at the global scale. Numerical experiments revealed that the mass effect of multiple elementary dipoles is expected to settle down into a behavior that qualitatively remains unchanged at different levels of a volume conductor. We summarize these findings as the principle of the conservation of the mass potential distributions.

First-Order Inconsistencies Caused by Rogue Trajectories

A theoretical requirement of the Interaction by Exchange with the Conditional Mean (IECM) micromixing model is that the mean concentration field produced by it must be consistent with the mean concentration field produced by a traditional Lagrangian stochastic (LS) marked particle model. We examine the violation of this requirement that occurs in a coupled LS-IECM model when unrealistically high particle velocities occur. No successful strategy was found to mitigate the effects of these rogue trajectories. It is our hope that this work will provide renewed impetus for investigation into rogue trajectories and methods to eliminate them from LS models.

Turbulent dispersion of a gas tracer in a nocturnal atmospheric flow

A Lagrangian turbulent dispersion model has been coupled with the Regional Atmospheric Modelling System (RAMS) to predict the transport, dispersion and concentration of a gas tracer released into a stable atmosphere (nocturnal drainage flow) of the Anderson Springs Valley near San Francisco, California. The mean wind velocities governed by the topography of the region and the surface layer fluxes of momentum and heat into the stable atmosphere, were calculated by the RAMS code. The atmospheric transport and dispersion of a gas tracer was simulated in non-homogeneous turbulence conditions by using a Lagrangian Stochastic Particle Model. A Moving Least Squares technique was used to calculate the tracer gas concentration. The vertical profiles of the mean wind speed and the wind direction, as well as the tracer gas concentration, were compared with the field measurements for the fourth night of the ASCOT experiment (the night between 19 and 20 September 1980) carried out in the Anderson Springs Valley, California, USA The computed profiles for wind direction and wind speed agreed well with the field data. Although at some locations the model predicted strong winds which transported the particle to larger distances downwind, the overall pattern of concentration distribution generated by the model compared quite well with the observations. It was concluded that the concentration patterns tend to follow the topography when the synoptic winds are weak.

A stochastic Lagrangian particle model and nonlinear filtering for three dimensional Euler flow with jumps

In this paper we will introduce a stochastic Lagrangian particle model with jumps for the three dimensional Euler ow and study the asso- ciated nonlinear ltering problem. We apply results from backward integro- dierential equation problem to prove uniqueness of solution to the Zakai equations.

Modelling Sequential Impact of Molten Droplets on a Solid Surface in Plasma Spray Process

Plasma spray deposition is one of the most important technologies available for producing the high-performance surfaces required by modern industry. In this process, powder of the coating material is fed into high-temperature plasma, which melts and accelerates the powder; the molten particles subsequently hit and solidify on the surface to be coated. To obtain good quality coating, the powder particle must be at least partially molten and hit the substrate with a high velocity. The flattening characteristics of the droplets impinging on a substrate are important determinants in governing the eventual quality of the plasma spray coating. Different codes have been developed in recent years to simulate the overall thermal spraying process, as well as the growth of the 3D coatings, in which entrained particles are modeled by stochastic particle models, fully coupled to the plasma flow. The present investigation was carried out to have an approach to systematize the atmospheric plasma spraying process in order to create a basis for numerically modeling the plasma dynamics, the coating formation mechanisms and to predict the particle thermo- kinetic state at impact.

A new puff modelling technique for short range dispersion applications

A puff model is proposed for short range dispersion applications. The model aims to reproduce the results of a stochastic Lagrangian particle model while avoiding to a large extent the noise which is often present in Lagrangian particle models. Part of the dispersion is modelled through the growth of the puffs while part is treated by a random motion of the puff centres. Comparisons with a Lagrangian particle model are presented for an idealised convective boundary layer. Also, the performance of the model in simulating the Kincaid experiment is described. The results are comparable with the best performing models for this dataset.

Bidirectional transport on a dynamic lattice

Bidirectional variants of stochastic many particle models for transport by molecular motors show a strong tendency to form macroscopic clusters on static lattices. Inspired by the fact that the microscopic tracks for molecular motors are dynamical, we study the influence of different types of lattice dynamics on stochastic bidirectional transport. We observe a transition toward efficient transport (corresponding to the dissolution of large clusters) controlled by the lattice dynamics.

Simulating the characteristic patterns of the dispersion during sunset PBL

A diffusion equation limit derived from the Langevin stochastic particle model is used to study the dispersion of scalars caused by an evolving turbulence in the transition process occurring during the sunset period. Therefore, the random displacement equation is employed to simulate the cross-wind concentrations of pollutants released from low and high point sources. Turbulence inputs are parameterized to describe in a continuous manner the dispersion effects produced by a decaying convective elevated and a shear-dominated stable surface turbulence. The simulation results show that for the initial stage of the sunset transition phenomenon, the pollutants are transported rapidly to the surface. On the other hand, for the sunset evolution advanced stages, the dispersion process happens in a deep stable boundary layer (SBL), in which the pollutants can travel long distances practically without reaching the surface. The major progress shown in this analysis is the description of the transport properties associated to decaying convective eddies in the residual layer (RL). The study shows that the diffusion effects associated to these decaying convective eddies strongly influence the dispersion of scalars during the sunset transition period.

A Detailed Model for the Sintering of Polydispersed Nanoparticle Agglomerates

In this study the coagulation, condensation, and sintering of nanoparticles is investigated using a stochastic particle model. Each stochastic particle consists of interacting polydisperse primary particles that are connected to each other. In the model sintering occurs between each individual pair of neighboring primary particles. This is important for particles in which the range of the size of the primary particles varies significantly. The sintering time is obtained from the viscous flow model. The model is solved using a stochastic particle algorithm. The particles are represented in a binary tree that contains the connectivity as well as the degree of sintering information. Particles are forme, coagulate, sinter, and experience condensation according to known rate laws. The particle binary tree, along with it the degree of sintering, is updated after each time step according to the rates of the different processes. The stochastic particle method uses the technique of fictitious jumps and linear proc...

Air pollution modelling using a Graphics Processing Unit with CUDA

The Graphics Processing Unit (GPU) is a powerful tool for parallel computing. In the past years the performance and capabilities of GPUs have increased, and the Compute Unified Device Architecture (CUDA) – a parallel computing architecture – has been developed by NVIDIA to utilize this performance in general purpose computations. Here we show for the first time a possible application of GPU for environmental studies serving as a basement for decision making strategies. A stochastic Lagrangian particle model has been developed on CUDA to estimate the transport and the transformation of the radionuclides from a single point source during an accidental release. Our results show that parallel implementation achieves typical acceleration values in the order of 80–120 times compared to CPU using a single-threaded implementation on a 2.33 GHz desktop computer. Only very small differences have been found between the results obtained from GPU and CPU simulations, which are comparable with the effect of stochastic transport phenomena in atmosphere. The relatively high speedup with no additional costs to maintain this parallel architecture could result in a wide usage of GPU for diversified environmental applications in the near future.

Difusão de contaminantes em condições de vento fraco empregando um modelo estocástico Lagrangeano

Um modelo estocástico Lagrangeano é utilizado para simular a dispersão e o transporte de contaminantes, sob condições estáveis, com velocidade do vento fraco, no experimento traçante realizado no Idaho National Engineering Laboratory (INEL). Neste trabalho é testada uma nova parametrização dos parâmetros que representam as freqüências associadas ao fenômeno de meandro do vento, presentes nas equações de Langevin para as componentes do vento horizontal. Esta nova parametrização é descrita em termos de uma quantidade adimensional que controla a freqüência de oscilação de meandro do vento e da escala de tempo, associada às estruturas coerentes em uma turbulência bem desenvolvida. As simulações demonstram que o modelo Lagrangeano considerado, incorporando esta nova parametrização, reproduz corretamente a difusão de escalares passivos em uma camada limite atmosférica estável com velocidade do vento fraco.

Convergence to weighted fractional Brownian sheets

We define weighted fractional Brownian sheets, which are a class of Gaussian random fields with four parameters that include fractional Brownian sheets as special cases, and we give some of their properties. We show that for certain values of the parameters the weighted fractional Brownian sheets are obtained as limits in law of occupation time fluctuations of a stochastic particle model. In contrast with some known approximations of fractional Brownian sheets which use a kernel in a Volterra type integral representation of fractional Brownian motion with respect to ordinary Brownian motion, our approximation does not make use of a kernel.

Stochastic and deterministic models for age-structured populations with genetically variable traits

Understanding how stochastic and non-linear deterministic processes interact is a major challenge in population dynamics theory. After a short review, we introduce a stochastic individual-centered particle model to describe the evolution in continuous time of a population with (continuous) age and trait structures. The individuals reproduce asexually, age, interact and die. The 'trait' is an individual heritable property (d-dimensional vector) that may influence birth and death rates and interactions between individuals, and vary by mutation. In a large population limit, the random process converges to the solution of a Gurtin-McCamy type PDE. We show that the random model has a long time behavior that differs from its deterministic limit. However, the results on the limiting PDE and large deviation techniques \textit{a la} Freidlin-Wentzell provide estimates of the extinction time and a better understanding of the long time behavior of the stochastic process. This has applications to the theory of adaptive dynamics used in evolutionary biology. We present simulations for two biological problems involving life-history trait evolution when body size is plastic and individual growth is taken into account.