谐振晶格的声模能量输运
Energy transport by acoustic modes of harmonic lattices
摘要：
本文研究了满足离散波动方程的三维标量晶格激发的大尺度演化。我们假设与波动方程的弹性耦合常数相关的色散关系是声学的，即，它在消失波矢量k=0附近具有|k|类型的奇点。为了推导出描述宏观能量输运的方程，我们引入了维格纳变换并改变变量，使得空间和时间尺度都是λ的量级。在连续极限中，通过将参数φ取为0，Wigner变换分解为三个不同的极限对象：弱极限的变换、H-测度和Wigner-测度。我们证明了这三个极限对象满足一组解耦的输运方程：重标初始数据的弱极限的波动方程，规则限制Wigner测度的色散输运方程，以及初始数据的H-测度极限集中到k=0的几何光学输运方程。我们结果的一个简单结果是具有声学色散关系的谐振晶格中能量输运的完整表征。
英文摘要：
We study the large scale evolution of a scalar lattice excitation which satisfies a discrete wave-equation in three dimensions. We assume that the dispersion relation associated to the elastic coupling constants of the wave-equation is acoustic, i.e., it has a singularity of the type |k| near the vanishing wave vector, k=0. To derive equations that describe the macroscopic energy transport we introduce the Wigner transform and change variables so that the spatial and temporal scales are of the order of epsilon. In the continuum limit, which is achieved by sending the parameter epsilon to 0, the Wigner transform disintegrates into three different limit objects: the transform of the weak limit, the H-measure and the Wigner-measure. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave-equation for the weak limit of the rescaled initial data, a dispersive transport equation for the regular limiting Wigner measure, and a geometric optics transport equation for the H-measure limit of the initial data concentrating to k=0. A simple consequence of our result is the complete characterization of energy transport in harmonic lattices with acoustic dispersion relations.
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