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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. 13921418. doi:10.1137/070699184
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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 weakL2 limit, an Hmeasure, 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 Hmeasure 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  

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:  00361410  
Official Date:  2008  
Dates: 


Volume:  Volume 40  
Number:  Number 4  
Number of Pages:  27  
Page Range:  pp. 13921418  
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 181/191 and Sp 181/192 (DFG) 
Data sourced from Thomson Reuters' Web of Knowledge
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