A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation

Bird's direct simulation Monte Carlo method for the Boltzmann equation is considered. The limit (as the number of particles tends to infinity) of the random empirical measures associated with the Bird algorithm is shown to be a deterministic measure-valued function satisfying an equation close (in a certain sense) to the Boltzmann equation. A Markov jump process is introduced, which is related to Bird's collision simulation procedure via a random time transformation. Convergence is established for the Markov process and the random time transformation. These results, together with some general properties concerning the convergence of random measures, make it possible to characterize the limiting behavior of the Bird algorithm.     

Quantum Boltzmann equation solved by Monte Carlo method for nano-scale semiconductor devices simulation

A two-dimensional (2D) full band self-consistent ensemble Monte Carlo (MC) method for solving the quantum Boltzmann equation, including collision broadening and quantum potential corrections, is developed to extend the MC method to the study of nano-scale semiconductor devices with obvious quantum mechanical (QM) effects. The quantum effects both in real space and momentum space in nano-scale semiconductor devices can be simulated. The effective mobility in the inversion layer of n and p channel MOSFET is simulated and compared with experimental data to verify this method. With this method 50nm ultra thin body silicon on insulator MOSFET are simulated. Results indicate that this method can be used to simulate the 2D QM effects in semiconductor devices including tunnelling effect.     

Monte Carlo solution of the Boltzmann equation via a discrete velocity model

A new discrete velocity scheme for solving the Boltzmann equation is described. Directly solving the Boltzmann equation is computationally expensive because, in addition to working in physical space, the nonlinear collision integral must also be evaluated in a velocity space. Collisions between each point in velocity space with all other points in velocity space must be considered in order to compute the collision integral most accurately, but this is expensive. The computational costs in the present method are reduced by randomly sampling a set of collision partners for each point in velocity space analogous to the Direct Simulation Monte Carlo (DSMC) method. The present method has been applied to a traveling 1D shock wave. The jump conditions across the shock wave match the Rankine–Hugoniot jump conditions. The internal shock wave structure was compared to DSMC solutions, and good agreement was found for Mach numbers ranging from 1.2 to 10. Since a coarse velocity discretization is required for efficient calculation, the effects of different velocity grid resolutions are examined. Additionally, the new scheme’s performance is compared to DSMC and it was found that upstream of the shock wave the new scheme performed nearly an order of magnitude faster than DSMC for the same upstream noise. The noise levels are comparable for the same computational effort downstream of the shock wave.     

Numerical simulation of metal evaporation based on the kinetic model equation and the direct simulation Monte Carlo method

Metal evaporation on the basis of the kinetic model equations(BGK and S-model) and the direct simulation Monte Carlo(DSMC) method was investigated computationally under the circumstances of collimators existing or not. Numerical data of distributions of number density, bulk velocity and temperature were reported over a wide range of evaporation rate.It was shown that these results reached a good agreement for the case of small evaporation rate, while the deviations became increasingly obvious with the increase of evaporation rate, especially when the collimators existed. Moreover, the deposition thickness over substrate obtained from the kinetic model equations were inaccurate even though the evaporation rate was small. All of the comparisons showed the reliability of the kinetic model equations, which require less computational cost at small evaporation rate and simple structure.     

A new multispin coding algorithm for Monte Carlo simulation of the Ising model

We present a new algorithm for Monte Carlo simulation of the Ising model. The usual serial architecture of a computer is exploited in a novel way, enabling parallel but independent calculations to be carried out on as many spins as there are bits in a computer word in each fundamental move. The algorithm enjoys a substantial increase in execution speed over more usual multispin coding algorithms. By its very nature, the algorithm constitutes a design for a special-purpose processor.     

A Monte Carlo algorithm for lattice ribbons

A lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions. The ribbon can be closed to form an object which is topologically either a cylinder or a Möbius band, depending on whether its surface is orientable or nonorientable. We describe a grand canonical Monte Carlo algorithm for generating a sample of these ribbons, prove that the associated Markov chain is ergodic, and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons.     

A Monte Carlo simulation of coagulation

A Monte Carlo simulation technique is described for the study of the coagulation of suspended particles. The method is computationally efficient since the particle trajectories are not used to determine coagulations. Instead, pairs of particles are assigned probabilities to coagulate and the evolution is computed as a stochastic Markov game. We also describe a simple analytic method to obtain the stationary distribution of sizes for the various mechanisms of relative particle motion. It is demonstrated that the simulation yields the correct stationary size distribution independent of initial condition.     

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.     

Fast vectorized algorithm for the Monte Carlo simulation of the random field Ising model

An algorithm for the simulation of the 3-dimensional random field Ising model with a binary distribution of the random fields is presented. It uses multi-spin coding and simulates 64 physically different systems simultaneously. On one processor of a Cray YMP it reaches a speed of 184 million spin updates per second. For smaller field strength we present a version of the algorithm that can perform 242 million spin updates per second on the same machine.     

A Monte Carlo simulation of random surfaces

We use a previously proposed stochastic process and carry out Monte Carlo simulations for two models of random surfaces defined on hypercubic lattices. Tests are carried out in five dimensions. The critical coupling constants are estimated for both models. One of the models allows also results on its entropy.     

Investigation of gate current in nano-scale MOSFETs by Monte Carlo solution of quantum Boltzmann equation

This paper investigates gate current through ultra-thin gate oxide of nano-scale metal oxide semiconductor field effect transistors (MOSFETs), using two-dimensional (2D) full-band self-consistent ensemble Monte Carlo method based on solving quantum Boltzmann equation. Direct tunnelling, Fowler--Nordheim tunnelling and thermionic emission currents have been taken into account for the calculation of total gate current. The 2D effect on the gate current is investigated by including the details of the energy distribution for electron tunnelling through the barrier. In order to investigate the properties of nano scale MOSFETs, it is necessary to simulate gate tunnelling current in 2D including non-equilibrium transport.     

MacPherson-Srolovitz晶粒长大速率方程的仿真验证

2007年MacPherson和Srolovitz联合推导出一个三维个体晶粒长大的准确速率方程,但并未给出实验或计算机仿真的验证.采用Potts模型Monte Carlo方法对该速率方程进行了大尺度仿真验证.结果表明,仿真数据与MacPherson-Srolovitz速率方程符合很好,从而初步证实了该速率方程,即三维晶粒长大速率是晶粒棱长和晶粒平均宽度的函数. 关键词： 三维晶粒长大 速率方程 Monte Carlo仿真     

Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation

The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past ...     

Beyond direct simulation Monte Carlo (DSMC) modelling of collision environments

ABSTRACT

Direct simulation Monte Carlo (DSMC) models have been successfully adopted and adapted to describe gas flows in a wide range of environments since the method was first introduced by Bird in the 1960s. We propose a new approach to modelling collisions between gas-phase particles in this work – operating in a similar way to the DSMC model, but with one key difference. Particles move in a mean field, generated by all previously propagated particles, which removes the requirement that all particles be propagated simultaneously. This yields a significant reduction in computation effort and lends itself to applications for which DSMC becomes intractable, such as when a species of interest is only a minor component of a large gas mixture.     

Monte Carlo simulation of mictomagnets

Monte Carlo simulations of a binary alloy with impurity concentrations between 20 and 45 at.% have been carried out. The proportion of large clusters relative to that of small clusters increases with the number of MC diffusion steps as well as impurity concentration. Magnetic susceptibility peaks become more prominent and occur at higher temperatures with increasing impurity concentration. The different peaks in the susceptibility and specific heat curves seem to correspond to different sized clusters. A freezing model would explain the observed behaviour with the large clusters freezing first and the small clusters contributing to susceptibility (specific heat) peaks at lower temperatures.Contribution No. 153 from the Solid State and Structural Chemistry Unit     

反应堆蒙特卡罗临界模拟中均匀裂变源算法的改进

在反应堆pin-by-pin精细建模及蒙特卡罗模拟计算研究中, 由于不同栅元的功率密度差异较大, 导致蒙特卡罗方法临界计算的样本在不同栅元之间的分配不均衡, 由此引起栅元内的各种计数的统计误差差异较大. 为使大部分栅元内计数的统计误差降至一个合理的水平, 单纯增加总样本已不是一个高效的解决方法. 通过在特定临界计算迭代算法的基础上改进并实现均匀裂变源算法的思想, 对大亚湾压水堆pin-by-pin模型取得了具有较高效率的数值结果. 本工作为具有自主知识产权的蒙特卡罗粒子输运模拟软件JMCT最终达到反应堆pin-by-pin模型(包括一系列国际基准模型)的模拟性能要求提供了一个有效的工具.     

A multi-phase Monte Carlo technique for simulation of radiative transfer

Evidence is presented which shows that the transfer of radiation in a set of homogeneous scattering atmospheres, characterized by different phase functions but the same value of ω_o, can be simultaneously simulated with Monte Carlo techniques in a single computer run. Furthermore, it is shown that the computation time increases very slowly with the number of phase functions processed.     

Hamiltonian evolution for the hybrid Monte Carlo algorithm

We discuss a class of reversible, discrete approximations to Hamilton's equations for use in the hybrid Monte Carlo algorithm and derive an asymptotic formula for the step-size-dependent errors arising from this family of approximations. For lattice QCD with Wilson fermions, we construct several different updates in which the effect of fermion vacuum polarization is given a longer time step than the gauge field's self-interaction. On a 4⁴ lattice, one of these algorithms with an optimal choice of step size is 30% to 40% faster than the standard leapfrog update with an optimal step size.     

Particle-in-cell plus direct simulation Monte Carlo (PIC-DSMC)approach for self-consistent plasma-gas simulations

The particle-in-cell (PIC) and direct simulation Monte Carlo (DSMC) approaches have been combined into a PIC-DSMC model for self-consistent simulations of low-temperature collisional plasmas and the background gas. This novel approach is based on the weighting collision simulation scheme allowing for disparate number densities and time scales of different species. The applicability of the developed algorithm is illustrated by simulations of one-dimensional direct current and two-dimensional magnetron sputtering discharges in argon. An appreciable effect of the energetic discharge species on the density, temperature, and flow field of the background gas shows the importance of the coupled plasma-gas simulation for such technologies as sputtering, dry etching, plasma enhanced vapor deposition, etc