Quantum Transport in a Low-Density Periodic Potential: Homogenisation via Homogeneous Flows

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 Erdös 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 four and higher. Our proof uses Floquet-Bloch theory, multi-variable 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.

arXiv comment45 pages, 1 figure

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Cited by in corpus (2)

• Quantum Transport in a Crystal With Short-Range Interactions: The Boltzmann-Grad Limit
• Derivation of the linear Boltzmann equation from the damped quantum Lorentz gas with a general scatterer configuration