Derivation of the linear Boltzmann equation from the damped quantum Lorentz gas with a general scatterer configuration

TL;DR

This paper rigorously derives the linear Boltzmann equation from a damped quantum Lorentz gas in the low-density limit, extending previous results to a broad class of scatterer configurations and emphasizing the role of damping.

Contribution

It generalizes the derivation of the linear Boltzmann equation to include damped quantum Lorentz gases with diverse scatterer arrangements, beyond the previously studied single or random configurations.

Findings

Damping is essential for convergence in the derivation.

The linear Boltzmann equation emerges for a wide class of configurations.

Without damping, the limiting behavior varies with configuration.

Abstract

It is a fundamental problem in mathematical physics to derive macroscopic transport equations from microscopic models. In this paper we derive the linear Boltzmann equation in the low-density limit of a damped quantum Lorentz gas for a large class of deterministic and random scatterer configurations. Previously this result was known only for the single-scatterer problem on the flat torus, and for uniformly random scatterer configurations where no damping is required. The damping is critical in establishing convergence -- in the absence of damping the limiting behaviour depends on the exact configuration under consideration, and indeed, the linear Boltzmann equation is not expected to appear for periodic and other highly ordered configurations.