Macroscopic Equations for Microscopic Dynamics in Periodic Crystals

Wednesday, April 13, 2005 - 2:30pm - 3:30pm
EE/CS 3-180
Alexander Mielke (Weierstraß-Institut für Angewandte Analysis und Stochastik (WIAS))
In infinite periodic lattices the solutions can be studied by Fourier analysis on the associated dual torus. However, in doing a limit procedure with vanishing atomic distance, one observes new phenoma which are usually studied by WKB mehtods. We show that similar results can be obtained under much weaker assumptions by using weak convergence methods.

First we show that linearized elastodynamics can be obtained by a gamma limit procedure which automatically produces the effective elastic tensor. Second we study the transport of energy in the lattice which occurs on quite different wave speeds as the macroscopic elastic waves. It is possible to derive a energy transport equation for a Wigner measure which depends on time, space and the wave vector on the dual torus.