Vol. 1, No. 4, 2019

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

Jory Griffin and Jens Marklof

Vol. 1 (2019), No. 4, 571–614
DOI: 10.2140/paa.2019.1.571

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.

theta functions, homogeneous dynamics, quantum transport
Mathematical Subject Classification 2010
Primary: 37A17, 82C10
Received: 2 November 2018
Revised: 20 March 2019
Accepted: 6 June 2019
Published: 12 October 2019
Jory Griffin
Department of Mathematics and Statistics
Queen’s University
Kingston, ON
Jens Marklof
School of Mathematics
University of Bristol
United Kingdom