Pure and Applied Analysis

Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows

Jory Griffin and Jens Marklof

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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 Erdős 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 4 and higher. Our proof uses Floquet–Bloch theory, multivariable 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.

Article information

Source
Pure Appl. Anal., Volume 1, Number 4 (2019), 571-614.

Dates
Received: 2 November 2018
Revised: 20 March 2019
Accepted: 6 June 2019
First available in Project Euclid: 29 October 2019

Permanent link to this document
https://projecteuclid.org/euclid.paa/1572314426

Digital Object Identifier
doi:10.2140/paa.2019.1.571

Mathematical Reviews number (MathSciNet)
MR4026550

Zentralblatt MATH identifier
07142202

Subjects
Primary: 37A17: Homogeneous flows [See also 22Fxx] 82C10: Quantum dynamics and nonequilibrium statistical mechanics (general)

Keywords
theta functions homogeneous dynamics quantum transport

Citation

Griffin, Jory; Marklof, Jens. Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows. Pure Appl. Anal. 1 (2019), no. 4, 571--614. doi:10.2140/paa.2019.1.571. https://projecteuclid.org/euclid.paa/1572314426


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