Karsten Matthies

From Particles to Boltzmann equations and (fractional) Diffusion

We will study to related particle models: the random Lorentz gas, where a tracer particle is moving between fixed random scatterer and the Rayleigh gas, where the tagged particle collides with moving scatterers that do not interact with themselves. First, in joint work with Raphael Winter, we study the timescale T(r) of the invariance principle, i.e. diffusive behaviour, for a Lorentz gas with particle size r and improve results obtained by Lutsko and Toth (2020). The work includes improved geometric recollision estimates and a discussion of the coupling of stochastic processes introduced in the original result. In the second part, the fractional diffusion equation is rigorously derived as a scaling limit from a deterministic Rayleigh gas, where
particles interact via short range potentials with support of size r and the background is distributed in space according to a Poisson process with intensity N and in velocity according to some fat-tailed distribution. As an intermediate step a linear Boltzmann equation is obtained in the Boltzmann-Grad limit as r tends to zero and N tends to infinity with N r^2 =c. The convergence of the empiric particle dynamics to the Boltzmann-type dynamics is shown using semigroup methods to describe probability measures on collision trees associated to physical trajectories in the case of a Rayleigh gas. The fractional diffusion equation is a hydrodynamic limit for times [0,T], where T and inverse mean free path c can both be chosen as some negative rational power r^{-k}. Based on joint work with Theodora Syntaka.