Pure and Applied Analysis
- Pure Appl. Anal.
- Volume 1, Number 4 (2019), 571-614.
Quantum transport in a low-density periodic potential: homogenisation via homogeneous flows
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.
Pure Appl. Anal., Volume 1, Number 4 (2019), 571-614.
Received: 2 November 2018
Revised: 20 March 2019
Accepted: 6 June 2019
First available in Project Euclid: 29 October 2019
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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