Harris, Lisa, Lukkarinen, Jani, Teufel, Stefan and Theil, Florian. (2008) Energy transport by acoustic modes of harmonic lattices. SIAM Journal on Mathematical Analysis, Vol.40 (No.4). pp. 1392-1418. ISSN 0036-1410

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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
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	SIAM Journal on Mathematical Analysis
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Date:	2008
Volume:	Vol.40
Number:	No.4
Number of Pages:	27
Page Range:	pp. 1392-1418
Identification Number:	10.1137/070699184
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
URI:	http://wrap.warwick.ac.uk/id/eprint/28554

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