Energy transport by acoustic modes of harmonic lattices
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 0036-1410Full text not available from this repository.
Official URL: http://dx.doi.org/10.1137/070699184
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|
|Number of Pages:||27|
|Page Range:||pp. 1392-1418|
|Access rights to Published version:||Restricted or Subscription Access|
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