Quantum transport in a crystal with short-range interactions: The Boltzmann-Grad limit

Jory Griffin; Jens Marklof

doi:10.1007/s10955-021-02797-z

Quantum transport in a crystal with short-range interactions: The Boltzmann-Grad limit

Jory Griffin, Jens Marklof

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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

Original language	English
Article number	16
Number of pages	46
Journal	Journal of Statistical Physics
Volume	184
Issue number	2
DOIs	https://doi.org/10.1007/s10955-021-02797-z
Publication status	Published - 26 Jul 2021

Bibliographical note

Funding Information:
Research supported by EPSRC Grant EP/S024948/1.

Publisher Copyright:
© 2021, The Author(s).

Keywords

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

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title = "Quantum transport in a crystal with short-range interactions: The Boltzmann-Grad limit",

abstract = "We study the macroscopic transport properties of the quantum Lorentzgas in a crystal with short-range potentials, and show that in the Boltzmann-Gradlimit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesisconcerning the statistical distribution of lattice points in thin domains, which isclosely related to the Berry-Tabor conjecture in quantum chaos",

keywords = "kinetic transport, Lorentz gas, Boltzmann equation, Floquet-Bloch theory, Berry-Tabor conjecture",

author = "Jory Griffin and Jens Marklof",

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AU - Marklof, Jens

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N2 - We study the macroscopic transport properties of the quantum Lorentzgas in a crystal with short-range potentials, and show that in the Boltzmann-Gradlimit the quantum dynamics converges to a random flight process which is not compatible with the linear Boltzmann equation. Our derivation relies on a hypothesisconcerning the statistical distribution of lattice points in thin domains, which isclosely related to the Berry-Tabor conjecture in quantum chaos

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

KW - kinetic transport

KW - Lorentz gas

KW - Boltzmann equation

KW - Floquet-Bloch theory

KW - Berry-Tabor conjecture

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DO - 10.1007/s10955-021-02797-z

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JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

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M1 - 16

ER -

Quantum transport in a crystal with short-range interactions: The Boltzmann-Grad limit

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Keywords

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