Communications in Mathematical Physics On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III
Communications in Mathematical Physics On the Distribution of Free Path Lengths for the Periodic Lorentz Gas III
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发表时间：
2003
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通讯作者：
E. Caglioti;F. Golse
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作者：
E. Caglioti;F. Golse
对于r∈(0,1)，令Zr = {x∈R2 | dist(x,Z2) > r/2}，并定义τr (x, v) = inf{t > 0 | x + tv∈∂Zr}。设r(t)为τr (x, v)≥t，且x和v分别均匀分布于Zr和S1的概率。本文证明了lim sup→0+ 1 | ln |∫1/4 r (t r) dr r = 2 π2t +O (1 t2), lim inf→0+ 1 | ln |∫1/4 r (t r) dr r = 2 π2t +O (1 t2), t→+∞。这个结果改进了Bourgain-Golse-Wennberg [common]中r的界。数学。物理学报，1999,19(5):589 - 598。我们还讨论了这一结果在动力学理论中的应用。1. 问题及主要结果说明周期性洛伦兹气体。设r∈(0,1 2)，定义Zr = {x∈R2 | dist(x,Z2) > r/2}。（1.1）考虑一个在Zr内以1速度运动的点粒子，每次遇到Zr边界时都发生镜面反射。这样的动力系统被称为“周期性二维洛伦兹气体”。（事实上，洛伦兹使用了动力学理论的方法来描述金属中电子的运动，就像点粒子在金属中原子的晶体结构上弹跳的无碰撞气体一样）。从x∈Zr开始，向v∈S1方向的“自由路径长度”（或“（正向）退出时间”）定义为τr (x, v) = inf{t > 0 | x + tv∈∂Zr}, (x, v)∈Zr × S1。（1.2） [E. Caglioti， F. Golse .
For r ∈ (0, 1), let Zr = {x ∈ R2 | dist(x,Z2) > r/2} and define τr (x, v) = inf{t > 0 | x + tv ∈ ∂Zr}. Let r(t) be the probability that τr (x, v) ≥ t for x and v uniformly distributed in Zr and S1 respectively. We prove in this paper that lim sup →0+ 1 | ln | ∫ 1/4 r ( t r ) dr r = 2 π2t +O ( 1 t2 ) , lim inf →0+ 1 | ln | ∫ 1/4 r ( t r ) dr r = 2 π2t +O ( 1 t2 ) as t → +∞. This result improves upon the bounds on r in Bourgain-Golse-Wennberg [Commun. Math. Phys. 190, 491–508 (1998)]. We also discuss the applications of this result in the context of kinetic theory. 1. Statement of the Problem and Main Results 1.1. The periodic Lorentz gas. Let r ∈ (0, 1 2 ) and define Zr = {x ∈ R2 | dist(x,Z2) > r/2}. (1.1) Consider a point particle moving at speed 1 inside Zr and being specularly reflected each time it meets the boundary of Zr . Such a dynamical system is referred to as “a periodic, two-dimensional Lorentz gas”. (Indeed, Lorentz used the methods of kinetic theory to describe the motion of electrons in a metal as that of a collisionless gas of point particles bouncing on the crystalline structure of atoms in the metal [13]). The “free path length” (or “(forward) exit time”) starting from x ∈ Zr in the direction v ∈ S1” is defined as τr (x, v) = inf{t > 0 | x + tv ∈ ∂Zr}, (x, v) ∈ Zr × S1. (1.2) 200 E. Caglioti, F. Golse