Abstract
We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance ɛ 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 methods.
Our aim is to derive macroscopic continuum limit equations for ɛ →0. For the weak limit of displacements and velocities we obtain the equation of linear elastodynamics, where the elasticity tensor is obtained as a Γ-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 a comparison to Whitham's modulation equation.
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Mielke, A. Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms. Arch Rational Mech Anal 181, 401–448 (2006). https://doi.org/10.1007/s00205-005-0405-2
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DOI: https://doi.org/10.1007/s00205-005-0405-2