Weak coupling limit for a tracer particle with rapid scatterings by light particles with point interactions
The primary result of this dissertation is a rigorous analysis of a scattering process for a test particle in the weak coupling limit. The reduced dynamics of the scattering process describes the free dynamics of a particle in 1 or 3 dimensions interlaced with scatterings with particles whose interaction is given by a point potential; the limiting dynamics describes a particle in an electro-magnetic field with an idealized non-classical noise. The weak coupling limit arises by taking the mass ratio λ = [special characters omitted] of a single scattering particle to the test particle to be near zero while the frequency of collisions increases proportionally to [special characters omitted]. The limiting dynamics is accurate up to second order in λ and the strength of both the vector potential of the induced electromagnetic field and the white noise is proportional to λ. We conjecture that after a second limit corresponding to a gas reservoir in an equilibrium state, the dynamics converge to a form studied for decoherence by Gallis and Fleming in 1990. We conclude by discussing decoherence related quantities of these limiting equilibrium dynamics.