精化中心极限定理与Lorentz气体Levy行走的无穷密度尾
Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk
摘要：
我们考虑点粒子与硬核弹性散射体的周期性阵列碰撞，其中自由飞行的长度是无限的（无限视界洛伦兹气体，LG）。Bleher中心极限定理（CLT）指出，粒子位移除以$\sqrt{t\ln t}$的分布在无限时间$t$的极限下是高斯分布。然而，最近有人强调，缓慢的收敛使这一结果无法观察到。使用LG的Lévy walk模型（LW），提出了使用重新缩放的Lambert函数而不是$\sqrt{t\ln t}$提供了快速收敛的、可观察的CLT，这被LG模拟所证实。我们在这里证明，这个结果可以简化为一个混合CLT的比例因子结合正常和异常扩散。对于狭窄的无限走廊，粒子在长时间内服从通常的正常扩散，这解释了以前的数值观察。在小散射体的相反限制中，Bleher CLT给出了良好的指导。在中间情况下，适用混合的CLT。高斯峰确定阶数小于2的矩。相比之下，CLT只给出了一半的坐标色散。丢失的一半色散和阶数高于2的矩由我们在这里导出的分布的尾部（无穷大密度）描述。尾巴是沿着无限的走廊支撑的，并由非常长的飞行形成，其持续时间与整个观察时间相当。从尾部计算的矩通过从定义为自上次碰撞以来经过的时间的向后递归时间的统计直接计算第四矩来确认。这就完成了LW模型的求解，可以与LG模型进行充分的比较。
英文摘要：
We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by $\sqrt{t\ln t}$ is Gaussian in the limit of infinite time $t$. However it was stressed recently that the slow convergence makes this result unobservable. Using a Lévy walk model (LW) of the LG, it was proposed that the use of a rescaled Lambert function instead of $\sqrt{t\ln t}$ provides a fast convergent, observable CLT, which was confirmed by the LG simulations. We demonstrate here that this result can simplified to a mixed CLT where the scaling factor combines normal and anomalous diffusions. For narrow infinite corridors the particle for long time obeys the usual normal diffusion, which explains the previous numerical observations. In the opposite limit of small scatterers the Bleher CLT gives a good guiding. In the intermediate cases the mixed CLT applies. The Gaussian peak determines moments of order smaller than two. In contrast, the CLT gives only half the coordinate dispersion. The missing half of the dispersion and also moments of order higher than two are described by the distribution's tail (the infinite density) which we derive here. The tail is supported along the infinite corridors and formed by anomalously long flights whose duration is comparable with the whole time of observation. The moments' calculation from the tail is confirmed by direct calculation of the fourth moment from the statistics of the backward recurrence time defined as time that elapsed since the last collision. This completes the solution of the LW model allowing full comparison with the LG.
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