Name
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. |
Year | DOI | Venue |
|---|---|---|
2008 | 10.1137/070699184 | SIAM JOURNAL ON MATHEMATICAL ANALYSIS |
Keywords | Field | DocType |
microlocal analysis,multiscale methods,homogenization,discrete wave equations | Coupling constant,Dispersion relation,Lattice (order),Mathematical analysis,Quantum mechanics,Scalar (physics),Singularity,Wave equation,Scaling,Mathematics,Wave vector | Journal |
Volume | Issue | ISSN |
40 | 4 | 0036-1410 |
Citations | PageRank | References |
0 | 0.34 | 0 |
Authors | ||
4 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Lisa Harris | 1 | 26 | 7.31 |
| Jani Lukkarinen | 2 | 0 | 0.34 |
| Stefan Teufel | 3 | 0 | 0.68 |
| Florian Theil | 4 | 14 | 5.66 |