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Center for Nonlinear Studies |
In the first part of the talk some sketch of the history of the subject is given. Starting from early work by Leontovich (1935) and Kac (1956), the development of stochastic models for the Boltzmann equation is discussed. These models are based on systems of particles imitating the behaviour of the gas molecules in a probabilistic way. A basic issue is to prove rigorously the convergence of the system (when the number of particles increases) to the solution of the equation in an appropriate sense. The second part of the talk is devoted to algorithmic and numerical aspects. The application of stochastic models for numerical purposes in rarefied gas dynamics goes back to Bird (1963). The so-called direct simulation Monte Carlo algorithm and its relation to the Leontovich model are discussed. Convergence results are briefly mentioned. In the third part of the talk some recent developments are presented. Some generalizations of the Boltzmann equation and the corresponding stochastic models are considered. They cover, for example, inelastic collisions (rarefied granular gases) and coagulation. A stochastic algorithm for the Uehling-Uhlenbeck-Boltzmann equation is discussed. Finally (as time permits) some result concerning the approximation of the steady Boltzmann equation is mentioned.