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### Energy transport by acoustic modes of harmonic lattices

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Harris, Lisa C., Lukkarinen, Jani, Teufel, Stefan, 1970- and Theil, Florian.
(2008)
*Energy transport by acoustic modes of harmonic lattices.*
SIAM Journal on Mathematical Analysis, Volume 40
(Number 4).
pp. 1392-1418.
ISSN 0036-1410

**Full text not available from this repository.**

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 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 |

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: | 0036-1410 |

Official Date: | 2008 |

Volume: | Volume 40 |

Number: | Number 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 |

Funder: | Deutsche Forschungsgemeinschaft (DFG), Suomen Akatemia [Academy of Finland] |

Grant number: | Sp 181/19-1 and Sp 181/19-2 (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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