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

Abstract

We consider Markov jump processes, Continuous Time Monte Carlo methods [A. I. Khisamutdinov, Simulation statistical modelling of the kinetic equation of rarefied gases. Dokl. Akad. Nauk SSSR (1988) 302, 75–79 (in Russian).] 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.