Macroscopic behavior of microscopic oscillations in harmonic lattices via Wigner--Husimi transforms
- Mielke, Alexander
2010 Mathematics Subject Classification
- 37K60 70F45 74Q10 74A25
- Harmonic lattice, discrete microscopic systems, macroscopic continuum limit, Gamma-limit, Wigner measure, Husimi transform
We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance epsilon tending to 0. We allow for general polyatomic crystals but assume exact periodicity such that the system can be solved in principle by Fourier transform and linear algebra. Our aim is to derive macroscopic continuum limit equations for epsilon --> 0. For the weak limit of displacements and velocities we find the equation of linear elastodynamics, where the elasticity tensor is obtained as a Gamma-limit. The weak limit of the local energy density can be described by generalizations of the Wigner-Husimi measure which satisfies a transport equation on the product of physical space and Fourier space. The concepts are illustrated via several examples and via a comparison to Whitham's modulation equation.
- Arch. Ration. Mech. Anal., 181 (2006) pp. 401--448.