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2019 Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows
Jory Griffin, Jens Marklof
Pure Appl. Anal. 1(4): 571-614 (2019). DOI: 10.2140/paa.2019.1.571

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.

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Jory Griffin. Jens Marklof. "Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows." Pure Appl. Anal. 1 (4) 571 - 614, 2019. https://doi.org/10.2140/paa.2019.1.571

Information

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

zbMATH: 07142202
MathSciNet: MR4026550
Digital Object Identifier: 10.2140/paa.2019.1.571

Subjects:
Primary: 37A17 , 82C10

Keywords: homogeneous dynamics , quantum transport , Theta functions

Rights: Copyright © 2019 Mathematical Sciences Publishers

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