Quantum Transport in a Crystal With Short-Range Interactions: The Boltzmann-Grad Limit

TL;DR

This paper investigates quantum transport in a crystal with short-range interactions, demonstrating that in the Boltzmann-Grad limit, the quantum dynamics converges to a non-Boltzmann random flight process, linking quantum chaos conjectures.

Contribution

It introduces a novel analysis showing quantum dynamics converges to a process incompatible with the linear Boltzmann equation, relying on a new hypothesis related to lattice point distributions.

Findings

Quantum dynamics converges to a non-Boltzmann process.

The analysis connects quantum transport to lattice point distribution hypotheses.

The results challenge the applicability of the linear Boltzmann equation in this context.

Abstract

We study the macroscopic transport properties of the quantum Lorentz gas in a crystal with short-range potentials, and show that in the Boltzmann-Grad limit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesis concerning the statistical distribution of lattice points in thin domains, which is closely related to the Berry-Tabor conjecture in quantum chaos.