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Energy transport by acoustic modes of harmonic lattices
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Harris, Lisa C., Lukkarinen, Jani, Teufel, Stefan and Theil, Florian (2008) Energy transport by acoustic modes of harmonic lattices. SIAM Journal on Mathematical Analysis, Volume 40 (Number 4). pp. 1392-1418. doi:10.1137/070699184 ISSN 0036-1410.
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Official URL: http://dx.doi.org/10.1137/070699184
Abstract
We study the large scale evolution of a scalar lattice excitation u which satisfies a discrete wave equation in three dimensions, u(t)(gamma) = -Sigma gamma' alpha(gamma - gamma') u(t)(gamma'), where gamma, gamma' epsilon Z(3) are lattice sites. We assume that the dispersion relation. associated to the elastic coupling constants alpha(gamma - gamma') is acoustic; i. e., it has a singularity of the type |k| near the vanishing wave vector, k = 0. To derive equations describing the macroscopic energy transport, we employ a related multiscale Wigner transform and a scale parameter epsilon > 0. The spatial and temporal scales of the Wigner transform are related to the corresponding lattice parameters via a scaling by e. In the continuum limit, which is achieved by sending the parameter e to 0, the Wigner transform disintegrates into three different limit objects: the Wigner transform of a rescaled weak-L-2 limit, an H-measure, and a Wigner measure. The first two provide the finer resolution of the energy concentrating at k = 0 so that a set of closed evolution equations may arise. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave equation for the weak limit, a geometric optics transport equation for the H-measure limit, and a dispersive transport equation for the standard limiting Wigner measure. This yields a complete characterization of macroscopic energy transport in harmonic lattices with regular acoustic dispersion relations.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Library of Congress Subject Headings (LCSH): | Heat -- Transmission, Microlocal analysis, Multiscale modeling, Homogenization (Differential equations), Mass transfer, Lattice theory | ||||
Journal or Publication Title: | SIAM Journal on Mathematical Analysis | ||||
Publisher: | Society for Industrial and Applied Mathematics | ||||
ISSN: | 0036-1410 | ||||
Official Date: | 2008 | ||||
Dates: |
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Volume: | Volume 40 | ||||
Number: | Number 4 | ||||
Number of Pages: | 27 | ||||
Page Range: | pp. 1392-1418 | ||||
DOI: | 10.1137/070699184 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access | ||||
Funder: | Deutsche Forschungsgemeinschaft (DFG), Suomen Akatemia [Academy of Finland] | ||||
Grant number: | Sp 181/19-1 and Sp 181/19-2 (DFG) |
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