The derivation of the Boltzmann equation from a particle model of a gas is currently a major area of research in mathematical physics. The standard approach to this problem is to study the BBGKY hierarchy, a system of equations that describe the distribution of the particles. A new method has recently been developed to tackle this problem by studying the probability of observing a specific history of events. We further develop this method to derive the linear Boltzmann equation in the Boltzmann-Grad scaling from two similar Rayleigh gas hard-sphere particle models. In both models the initial distribution of the particles is random and their evolution is deterministic. Validity is shown up to arbitrarily large times and with only moderate moment assumptions on the non-equilibrium initial data. The first model considers a Rayleigh gas whereby one tagged particle collides with a large number of background particles, which have no self interaction. The initial distribution of the background particles is assumed to be spatially homogeneous and at a collision between a background particle and the tagged particle only the tagged particle changes velocity.In the second model we make two changes: we allow the background particles to have a spatially non-homogeneous initial data and we assume that at collision both the tagged particle and background particle change velocity. The proof for each model follows the same general method, where we consider two evolution equations, the idealised and the empirical, on all possible collision histories. It is shown by a semigroup approach that there exists a solution to the idealised equation and that this solution is related to the solution of the linear Boltzmann equation. It is then shown that under the particle dynamics the distribution of collision histories solves the empirical equation. Convergence is shown by comparing the idealised and empirical equations.
|Date of Award||23 Oct 2017|
|Supervisor||Karsten Matthies (Supervisor)|