Armstead Douglas N, Hunt Brian R, Ott Edward
Department of Physics and Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20904, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 1):021110. doi: 10.1103/PhysRevE.67.021110. Epub 2003 Feb 26.
We consider the long time dependence for the moments of displacement <|r|(q)> of infinite horizon billiards, given a bounded initial distribution of particles. For a variety of billiard models we find <|r|(q)> approximately t(gamma(q)) (up to factors of ln t). The time exponent, gamma(q), is piecewise linear and equal to q/2 for q<2 and q-1 for q>2. We discuss the lack of dependence of this result on the initial distribution of particles and resolve apparent discrepancies between this time dependence and a prior result. The lack of dependence on initial distribution follows from a remarkable scaling result that we obtain for the time evolution of the distribution function of the angle of a particle's velocity vector.
给定粒子的有界初始分布，我们考虑无限视界台球位移矩<|r|(q)>的长时间依赖性。对于各种台球模型，我们发现<|r|(q)>近似为t(gamma(q))（直至ln t的因子）。时间指数gamma(q)是分段线性的，对于q<2等于q/2，对于q>2等于q - 1。我们讨论了该结果与粒子初始分布无关的情况，并解决了这种时间依赖性与先前结果之间明显的差异。对初始分布的无关性源于我们为粒子速度矢量角度的分布函数的时间演化所获得的一个显著的标度结果。
