4.4 Article

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

Journal

JOURNAL OF STATISTICAL PHYSICS
Volume 184, Issue 2, Pages -

Publisher

SPRINGER
DOI: 10.1007/s10955-021-02797-z

Keywords

Kinetic transport; Lorentz gas; Boltzmann equation; Floquet-Bloch theory; Berry-Tabor conjecture

Funding

  1. EPSRC [EP/S024948/1]
  2. EPSRC [EP/S024948/1] Funding Source: UKRI

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The study reveals that in a crystal with short-range potentials, the macroscopic transport properties of the quantum Lorentz gas converge to a random flight process in the Boltzmann-Grad limit, which is not compatible with the linear Boltzmann equation. This derivation is based on a hypothesis about the statistical distribution of lattice points in thin domains, closely related to the Berry-Tabor conjecture in quantum chaos.
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.

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