Quantum Transport in a Low-Density Periodic Potential: Homogenisation via Homogeneous Flows

TL;DR

This paper investigates the quantum dynamics of particles in a periodic potential, demonstrating convergence to the linear Boltzmann equation in a specific limit and discussing the limitations of this approximation at higher orders.

Contribution

It provides a rigorous analysis of quantum transport in periodic potentials, extending homogenisation theory with new mathematical techniques and highlighting the limits of the Boltzmann equation approximation.

Findings

Convergence to the linear Boltzmann equation up to second order in the coupling constant.

Identification of potential failure of the Boltzmann equation at fourth order and beyond.

Use of Floquet-Bloch theory and equidistribution theorems in the proof.

Abstract

We show that the time evolution of a quantum wavepacket in a periodic potential converges in a combined high-frequency/Boltzmann-Grad limit, up to second order in the coupling constant, to terms that are compatible with the linear Boltzmann equation. This complements results of Eng and Erd\"os for low-density random potentials, where convergence to the linear Boltzmann equation is proved in all orders. We conjecture, however, that the linear Boltzmann equation fails in the periodic setting for terms of order four and higher. Our proof uses Floquet-Bloch theory, multi-variable theta series and equidistribution theorems for homogeneous flows. Compared with other scaling limits traditionally considered in homogenisation theory, the Boltzmann-Grad limit requires control of the quantum dynamics for longer times, which are inversely proportional to the total scattering cross section of the single-site potential.