This article is available for purchase or by subscription. See below.
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.
|
PDF Access Denied
We have not been able to recognize your IP address
3.145.115.195
as that of a subscriber to this journal.
Online access to the content of recent issues is by
subscription, or purchase of single articles.
Please contact your institution's librarian suggesting a subscription, for example by using our
journal-recommendation form.
Or, visit our
subscription page
for instructions on purchasing a subscription.
You may also contact us at
contact@msp.org
or by using our
contact form.
Or, you may purchase this single article for
USD 40.00:
Keywords
theta functions, homogeneous dynamics, quantum transport
|
Mathematical Subject Classification 2010
Primary: 37A17, 82C10
|
Milestones
Received: 2 November 2018
Revised: 20 March 2019
Accepted: 6 June 2019
Published: 12 October 2019
|
|