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Derivation of a kinetic model from a stochastic particle system
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We study a stochastic lattice particle system with exclusion prin-ciple. A kinetic equation and its diffusion limit are formally derived from the Monte Carlo dynamics. This derivation is investigated analytically and numerically and compared with the classical hydrodynamic limit of the sto-chastic exclusion process. Numerical results are presented for different values of jump probabilities.
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... The rigorous derivation of mean-field and fluid dynamic models from the Boltzmann equation is a deeply fascinating issue. Without intending to review all the literature on the topic, we indicate [12,20,21,23,33] as representative works of the hierarchy of scales described by kinetic modelling. ...
... Interestingly, owing to (17) the large time trends of g(t, v) and f (t, v; θ) are well approximated by those of g(τ, v) and f (τ, v; θ) satisfying (20), (21). (18), (19) reveals that, since η has zero mean, the evolution of m gγ (τ ), m fγ (τ ; θ) coincides for every γ > 0, hence also in the limit γ → 0 + , with the one predicted by the interaction models (3), (4). ...
... Owing to (23), the function P featuring in (20), (21) is P (v, w; θ) = (cos θ |cos θ| p − 1) v. In particular, recalling that θ ∼ U(0, 2π), ...
In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.
This paper deals with the modeling of production processes in automotive industries by models based on partial differential equations. The basic idea consists on the derivation of kinetic equations to model production flow on an assembly line. Numerical results based on data of an assembly plant are presented. The work implements a recent discussion for general flow on unstructured networks.
We review in this article the current theoretical understanding of collective and single particle diffusion on surfaces and how it relates to the existing experimental data. We begin with a brief survey of the experimental techniques that have been employed for the measurement of the surface diffusion coefficients. This is followed by a section on the basic concepts involved in this field. In particular, we wish to clarify the relation between jump or exchange motion on microscopic length scales, and the diffusion coefficients which can be defined properly only in the long length and time scales. The central role in this is played by the memory effects. We also discuss the concept of diffusion under nonequilibrium conditions. In the third section, a variety of different theoretical approaches that have been employed in studying surface diffusion such as first principles calculations, transition state theory, the Langevin equation, Monte Carlo and molecular dynamics simulations, and path integral formalism are presented. These first three sections form an introduction to the field of surface diffusion. Section 4 contains subsections that discuss surface diffusion for various systems which have been investigated both experimentally and theoretically. The focus here is not so much on specific systems but rather on important issues concerning diffusion measurements or calculations. Examples include the influence of steps, diffusion in systems undergoing phase transitions, and the role of correlation and memory effects. Obviously, the choice of topics here reflects the interest and expertise of the authors and is by no means exhaustive. Nevertheless, these topics form a collection of issues that are under active investigation, with many important open questions remaining.
The methodology of computer simulations of crystal growth is described. Two main methods, kinetic Monte Carlo and molecular dynamics, are discussed. The principle of kinetic Monte Carlo simulations is explained in detail, including recent developments of algorithms. Particular attention is paid to approximations which are made in the construction of discrete growth models. Applications of the Monte Carlo method for three different kind of problems: kinetic roughening, near equilibrium growth, and far-from-equilibrium molecular beam epitaxy growth are presented together with examples of representative results. Possibilities of employing molecular dynamics simulations are discussed as well, and examples of results are also given. The range of applicability of different methods on present-day computers is evaluated.
Scales.- Outline.- I Classical Particles.- 1. Dynamics.- 1.1 Newtonian Dynamics.- 1.2 Boundary Conditions.- 1.3 Dynamics of Infinitely Many Particles.- 2. States of Equilibrium and Local Equilibrium.- 2.1 Equilibrium Measures, Correlation Functions.- 2.2 The Infinite Volume Limit.- 2.3 Local Equilibrium States.- 2.4 Local Stationarity.- 2.5 The Static Continuum Limit.- 3. The Hydrodynamic Limit.- 3.1 Propagation of Local Equilibrium.- 3.2 Hydrodynamic Equations.- 3.3 The Hard Rod Fluid.- 3.4 Steady States.- 4. Low Density Limit: The Boltzmann Equation.- 4.1 Low Density (Boltzmann-Grad) Limit.- 4.2 BBGKY Hierarchy for Hard Spheres and Collision Histories.- 4.3 Convergence of the Scaled Correlation Functions.- 4.4 The Boltzmann Hierarchy.- 4.5 Time Reversal.- 4.6 Law of Large Numbers, Local Poisson.- 4.7 The H-Function.- 4.8 Extensions.- 5. The Vlasov Equation.- 6. The Landau Equation.- 7. Time Correlations and Fluctuations.- 7.1 Fluctuation Fields.- 7.2 The Green-Kubo Formula.- 7.3 Transport for the Hard Rod Fluid.- 7.4 The Fluctuating Boltzmann Equation.- 7.5 The Fluctuating Vlasov Equation.- 8. Dynamics of a Tracer Particle.- 8.1 Brownian Particle in a Fluid.- 8.2 The Stationary Velocity Process.- 8.3 Brownian Motion (Hydrodynamic) Limit.- 8.4 Large Mass Limit.- 8.5 Weak Coupling Limit.- 8.6 Low Density Limit.- 8.7 Mean Field Limit.- 8.8 External Forces and the Einstein Relation.- 8.9 Self-Diffusion.- 8.10 Corrections to Markovian Limits.- 9. The Role of Probability, Irreversibility.- II Stochastic Lattice Gases.- 1. Lattice Gases with Hard Core Exclusion.- 1.1 Dynamics.- 1.2 Stochastic Reversibility.- 1.3 Invariant Measures, Ergodicity, Domains of Attraction.- 1.4 Driven Lattice Gases.- 1.5 Standard Models.- 2. Equilibrium Fluctuations.- 2.1 Density Correlations and Bulk Diffusion.- 2.2 The Green-Kubo Formula.- 2.3 Currents.- 2.4 The Gradient Condition.- 2.5 Linear Response, Conductivity.- 2.6 Steady State Transport.- 2.7 State of Minimal Entropy Production.- 2.8 Bounds on the Conductivity.- 2.9 The Field of Density Fluctuations.- 2.10 Scaling Limit for the Density Fluctuation Field (Proof).- 2.11 Critical Dynamics.- 3. Nonequilibrium Dynamics for Reversible Lattice Gases.- 3.1 The Nonlinear Diffusion Equation.- 3.2 Hydrodynamic Limit (Proof).- 3.3 Low Temperatures.- 3.4 Weakly Driven Lattice Gases.- 3.5 Nonequilibrium Fluctuations.- 3.6 Local Equilibrium States and Minimal Entropy Production.- 3.7 Large Deviations.- 4. Nonequilibrium Dynamics of Driven Lattice Gases.- 4.1 Hyperbolic Equation of Conservation Type.- 4.2 Asymmetric Exclusion Dynamics.- 4.3 Fluctuation Theory.- 5. Beyond the Hydrodynamic Time Scale.- 5.1 Navier-Stokes Correction for Driven Lattice Gases.- 5.2 Local Structure of a Shock.- 5.2.1 Macroscopic Equation with Fluctuations.- 5.2.2 Shock in a Random Frame of Reference.- 5.2.3 Shock in Higher Dimensions.- 6. Tracer Dynamics.- 6.1 Two Component Systems.- 6.2 Tracer Diffusion.- 6.3 Convergence to Brownian Motion.- 6.4 Nearest Neighbor Jumps in One Dimension: The Case of Vanishing Self-Diffusion.- 7. Stochastic Models with a Single Conservation Law Other than Lattice Gases.- 7.1 Lattice Gases Without Hard Core/Zero Range Dynamics.- 7.2 Interacting Brownian Particles.- 7.3 Ginzburg-Landau Dynamics.- 8. Non-Hydrodynamic Limit Dynamics.- 8.1 Kinetic Limit.- 8.2 Mean Field Limit.- References.- List of Mathematical Symbols.
A general overview of methods of modeling thin film growth on an atomic scale is provided, focusing on molecular dynamics (MD), energy minimization, and Monte Carlo methods. We emphasize how different methods can be integrated to provide a more powerful predictive capability. We give examples of how one can use atomic-level information to predict nucleation rates and facet growth rates of Cu islands.
The influence of interactions between adsorbed particles on their diffusion constant is investigated by kinetic Ising models with independent nearest neighbour hops. This leads to expressions for the ratio of the diffusion constant at arbitrary coverage to its value at =0 as a function of the interaction energies relative to temperature. It is shown that under certain conditions this quantity obeys a particle-hole symmetry. Exact results in the whole range of densities are given in one dimension for nearest neighbour interaction. They already yield a qualitative agreement with experimental results and are also compared to corresponding numerical simulations. The introduction of a next nearest neighbour interaction is shown to produce drastic changes in the density dependence of the diffusion constant in some of the cases. A generalized quasichemical approximation and a virial expansion are made in two dimensions, leading to a better agreement with the measurements.
Monte Carlo methods are utilized as computational tools in many areas of chemical physics. In this paper, we present the theoretical basis for a dynamical Monte Carlo method in terms of the theory of Poisson processes. We show that if: (1) a "dynamical hierarchy" of transition probabilities is created which also satisfy the detailed-balance criterion; (2) time increments upon successful events are calculated appropriately; and (3) the effective independence of various events comprising the system can be achieved, then Monte Carlo methods may be utilized to simulate the Poisson process and both static and dynamic properties of model Hamiltonian systems may be obtained and interpreted consistently.
Monte Carlo models of crystal growth have contributed to the theoretical understanding of thin film deposition, and are now becoming available as tools to assist in device fabrication. Because they combine efficient computation and atomic-level detail, these models can be applied to a large number of crystallization phenomena. They have played a central role in the understanding of the surface roughening transition and its effect on crystal growth kinetics. In addition, columnar growth, vacancy and impurity trapping, and other growth phenomena that are closely related to atomic-level structure have been investigated by these simulations. In this chapter we review some of these applications and discuss MC modeling of sputter deposition based on materials parameters derived from first principles and molecular dynamics methods. We discuss models of deposition which include the atomic scale, but can also simulate film structure evolution on time scales of the order of hours. By the use of advanced computers and algorithms, we can now simulate systems large enough to exhibit clustered, columnar, and polycrystalline film structures. The event distribution is determined from molecular dynamics simulations, which can give diffusion rates, defect production, sputtering yields, and other information needed to match real materials. We discuss simulations of deposition into small vias and trenches, and their extension to the length scale of real devices through scaling relations.
We study a finite-difference discretization of an ill-posed nonlinear parabolic partial differential equation. The PDE is the one-dimensional version of a simplified two-dimensional model for the formation of shear bands via anti-plane shear of a granular medium. For the discretized initial value problem, we derive analytically, and observed numerically, a two-stage evolution leading to a steady-state: (i) an initial growth of grid-scale instabilities, and (ii) coarsening dynamics. Elaborating the second phase, at any fixed time the solution has a piecewise linear profile with a finite number of shear bands. In this coarsening phase, one shear band after another collapses until a steady-state with just one jump discontinuity is achieved. The amplitude of this steady-state shear band is derived analytically, but due to the ill-posedness of the underlying problem, its position exhibits sensitive dependence. Analyzing data from the simulations, we observe that the number of shear bands at time t decays like t−1/3. From this scaling law, we show that the time-scale of the coarsening phase in the evolution of this model for granular media critically depends on the discreteness of the model. Our analysis also has implications to related ill-posed nonlinear PDEs for the one-dimensional Perona–Malik equation in image processing and to models for clustering instabilities in granular materials.
A stable relaxation approximation for a transport equation with the diffusive scaling is developed. The relaxation approximation leads in the small mean free path limit to the higher order diffusion equation obtained from the asymptotic analysis of the transport equation. 1 Introduction In the present short communication we develop a relaxation approximation for a transport equation with the diffusive scaling. In [3] a relaxation approximation has been developed for kinetic equations with the Euler scaling. It has been used to stabilize higher order equations -- the Burnett equations -- obtained from the Chapman Enskog procedure. Here a relaxation approximation is developed for equations with the diffusion scaling. The relaxation approximation leads in the limit to a higher order approximation (O(ffl 4 )) of the transport equation. The approximations can be developed using an arbitrary number of moment equations. We consider systems with two and three equations. The procedure can be...
An asymptotic preserving numerical scheme (with respect to diffusion scalings) for a linear transport equation is investigated. The scheme is adopted from a class of schemes developed in [S. Jin, L. Pareschi, and G. Toscani, SIAM J. Numer. Anal., 38 (2000), pp. 913--936] and [A. Klar, SIAM J. Numer. Anal., 35 (1998), pp. 1073--1094]. Stability is proven uniformly in the mean free path under a CFL-type condition turning into a parabolic CFL condition in the diffusion limit.