Free Path Lengths in Quasicrystals
Free Path Lengths in Quasicrystals
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准晶体中的自由程长度
DOI：
10.1007/s00220-014-2011-3
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发表时间：
2013
影响因子：
2.4
通讯作者：
Andreas Strömbergsson
中科院分区：
文献类型：
--
作者：
J. Marklof;Andreas Strömbergsson
先前对洛伦兹气体中动力学输运的研究仅限于散射体随机分布的情况（例如，在空间泊松过程的点上）或欧几里得晶格的顶点上。在本文中，我们研究了非周期但强相关的准晶体散射结构。一个著名的例子是彭罗斯平铺的顶点集。我们的主要结果证明了自由路径长度的极限分布的存在性，从而回答了温伯格问题。该极限分布在高维晶格空间上以一定的随机变量为特征，与随机散射体结构的指数分布明显不同。这些证明的关键成分是齐次空间上的等分布定理，它是由Ratner的测度分类推导出来的。
Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of a Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner’s measure classification.