Collisional Models for a Quantum Particle in a Gas

In the last decades important steps has been made to understand the quantum behaviour of a test particle in a gas. However the most refined theoretical collisional models proposed so far seem not to be in agreement with each other. The open dabate about which model is the one that correctly describes a particle in a gas is a witness that the validity of these models is still unclear. A better understanding of the quantum behavior of a test particle in a gas is not only desirable, but would also help in understanding the non-classical process of decoherence, which is believed to be of crucial importance in the quantum-classical transition. The broad objective of this study is to analyze the validity of collisional models for the motion of a tracer particle affected by the presence of a background gas. We start with a critical review of the state of the art in quantum collisional models. Then, we study a very simple system, which is exactly solvable: A two-particle system interacting via a Dirac delta potential in one dimension. We analyze the interaction and estimate the collision time for Gaussian wave packets. Then, we focus on the main problem of this thesis: the dynamics of a test particle in a quantum gas. We first tackle it with an original technique that combines the Hartree variational method with stochastic calculus techniques. In this way we properly describe the non dissipative behavior of the test particle, and we gather interesting insight on the dissipative process. Eventually, we provide a microscopic derivation of the collisional dynamics for a test particle in a rarefied thermal bath. We, shows the limits of this approach, providing necessary conditions for the validity of collisional equation