Comparison of modes of convergence in a particle system related to the Boltzmann equation

Självständigt arbete på avancerad nivå (masterexamen) 20 poäng / 30 hp The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N -particle system. In the equilibrium case, they prove in [3] that the L 1 -distance between the density function of k particles in the N -particle process and the k -fold product of the solution to the stationary Boltzmann equation is of order 1/N . They do this in order to show that the N -particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity. This is different from the standard approach of describing convergence of an N -particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N -particle system as N tends to infinity assuming stationarity. The distribution of particles in a rarefied gas in a vessel can be described by the Boltzmann equation. As an approximation of the solution to this equation, Caprino, Pulvirenti and Wagner [3] constructed a random N -particle system. In the equilibrium case, they prove in [3] that the L 1 -distance between the density function of k particles in the N -particle process and the k -fold product of the solution to the stationary Boltzmann equation is of order 1/N . They do this in order to show that the N -particle system converges to the system described by the stationary Boltzmann equation as the number of particles tends to infinity. This is different from the standard approach of describing convergence of an N -particle system. Usually, convergence in distribution of random measures or weak convergence of measures over the space of probability measures is used. The purpose of the present thesis is to compare different modes of convergence of the N -particle system as N tends to infinity assuming stationarity. Självständigt arbete på avancerad nivå (masterexamen) 20 poäng / 30 hp