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On some properties of Markov processes and Monte Carlo methods for inhomogeneous Boltzmann equation

机译:非齐次玻尔兹曼方程的马尔可夫过程和蒙特卡罗方法的一些性质

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We consider Markov jump processes, Continuous Time Monte Carlo methods based on these processes, and the inhomogeneous smoothed Boltzmann equation. In the first place, integral forms of a system of master equations for the processes are constructed and general properties of the solution of the system are inspected. Secondly, we investigate unbiased simulation estimators for the computation of linear functionals on the phase density of the average number of particles; and we investigate the asymptotic behaviour of the variance of one of these estimators when the mean initial number density of particles increases. Third, we touch upon the convergence of the phase density to the solution to the Boltzmann equation. The paper considers the general case of a variable number of particles in the system and describes an example of the application of the methods. The results have relation to the known Direct Simulation Monte Carlo methods.
机译:我们考虑了马尔可夫跳过程,基于这些过程的连续时间蒙特卡洛方法以及不均匀的平滑玻尔兹曼方程。首先,构建用于过程的主方程系统的积分形式,并检查系统解的一般性质。其次,我们研究了无偏模拟估计量,用于计算平均粒子数相密度上的线性泛函。并且我们研究了当粒子的平均初始数量密度增加时,这些估计器之一的方差的渐近行为。第三,我们将相密度收敛到玻尔兹曼方程的解。本文考虑了系统中可变数量粒子的一般情况,并描述了该方法应用的示例。结果与已知的直接模拟蒙特卡洛方法有关。

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