Diffusive limit for a quantum linear Boltzmann dynamics

TL;DR

This paper proves that a quantum particle interacting with a dilute gas exhibits diffusive behavior, with its position distribution converging to a Gaussian under appropriate scaling, based on a quantum linear Boltzmann model.

Contribution

It establishes the diffusive limit for a quantum linear Boltzmann equation with hard-sphere scattering, demonstrating Gaussian convergence of the particle's position distribution.

Findings

Position distribution converges to a Gaussian under diffusive scaling.

The model uses a translation-covariant Lindblad equation for quantum dynamics.

Diffusive behavior is rigorously proven for the quantum Boltzmann model.

Abstract

In this article, I study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model I begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the gas particle scattering is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix that evolves according to a translation-covariant Lindblad equation. The main result is a proof that the particle's position distribution converges to a Gaussian under diffusive rescaling.