Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms
Macroscopic Behavior of Microscopic Oscillations in Harmonic Lattices via Wigner-Husimi Transforms
复制标题
通过维格纳-胡西米变换研究谐波晶格中微观振荡的宏观行为
DOI：
10.1007/s00205-005-0405-2
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发表时间：
2006
影响因子：
2.5
通讯作者：
A. Mielke
中科院分区：
文献类型：
--
作者：
A. Mielke
考虑无限调和格在格距趋于0的极限下的动力学问题。我们允许一般的多原子晶体，但假设精确的周期性，这样系统可以解决，原则上，通过傅里叶变换和线性代数方法。我们的目的是推导出宏观连续统极限方程。对于位移和速度的弱极限，我们得到了线性弹性动力学方程，其中弹性张量为Γ-limit。局部能量密度的弱极限可以用Wigner-Husimi测度的推广来描述，它满足物理空间与傅里叶空间乘积上的输运方程。通过几个例子和与惠瑟姆调制方程的比较说明了这些概念。
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.