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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. ISSN 00361410
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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 > 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  
Identification Number:  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)  
References:  [1] N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New 

URI:  http://wrap.warwick.ac.uk/id/eprint/28554 
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