The Lorentz process converges to a random flight process

The Lorentz process is the stochastic process defined by a particle moving, according to Newton's law of motion, through static scatterers distributed according to some probability measure in space. We consider the Boltzmann-Grad limit: The density of scatterers increases to infinity and at the same time the diameter of the scatterers decreases to zero in such a way that the mean free path of the particle is kept constant. We show that the Lorentz process converges in the weak*-topology of regular Borel measures on the paths space to some stochastic process. The limit process is Markovian if and only if the rescaled density of scatterers converges in probability to its mean. In that case the limit process is a (spatially inhomogeneous) random flight process.On leave of absence of Fachbereich Physik der Universität München. Work supported by a DFG fellowship