Lorentz气体的低密度极限：周期、非周期和随机
The low-density limit of the Lorentz gas: periodic, aperiodic and random
摘要：
洛伦兹气体是研究稀薄气体在物质中输运性质的最简单、最广泛使用的模型之一。它描述了一个云的非相互作用的点粒子在一个无限阵列的固定的球形散射体的动力学。在其概念提出一百多年后，理解动力学输运方程的性质仍然是一个重大挑战，该方程在低散射体密度极限（玻尔兹曼-格拉德极限）下控制宏观粒子动力学。洛伦兹认为这个方程应该是线性玻尔兹曼方程。这在加拉沃蒂、斯波恩、博尔德里吉尼、布尼莫维奇和西奈的三篇著名论文中得到了证实，他们假设散射体的分布是充分无序的。在强相关散射体配置（如晶体或准晶体）的情况下，我们现在理解了为什么线性玻尔兹曼方程失败，以及用什么来代替它。周期性洛伦兹气体的一个特别显著的特征是自由程长度分布的重尾，具有发散的二阶矩，以及在大时间的极限下的超扩散输运。
英文摘要：
The Lorentz gas is one of the simplest, most widely used models to study the transport properties of rarified gases in matter. It describes the dynamics of a cloud of non-interacting point particles in an infinite array of fixed spherical scatterers. More than one hundred years after its conception, it is still a major challenge to understand the nature of the kinetic transport equation that governs the macroscopic particle dynamics in the limit of low scatterer density (the Boltzmann-Grad limit). Lorentz suggested that this equation should be the linear Boltzmann equation. This was confirmed in three celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai, under the assumption that the distribution of scatterers is sufficiently disordered. In the case of strongly correlated scatterer configurations (such as crystals or quasicrystals), we now understand why the linear Boltzmann equation fails and what to substitute it with. A particularly striking feature of the periodic Lorentz gas is a heavy tail for the distribution of free path lengths, with a diverging second moment, and superdiffusive transport in the limit of large times.
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