关于敏感动力系统以及从表观决定过程向完全随机过程的过渡

本文详细分析了掷硬币的动力学过程,通过计算研究了最终位形(即正、反面)究竟如何敏感地依赖于初始条件以及造成这种敏感性的原因.结果也表明,随着硬币质心初始高度h的增加,最终位形对于初始参数(初始方位.初始角速度等)、桌面的能量吸收因子以及空气阻力系数等变得越来越敏感.如果我们在初始时刻保持“正面”向上.但允许某些其它参数有一个微小的变化范围,那么,当h小(与硬币半径相比)时.最终位形为“正面”的频率为1:当h非常大时,该频率接近于1/2.一个有趣的问题是:当h从零开始连续增加时,这个频率怎样从1连续地过渡到1/2?仔细计算表明,这个“过渡”与从层流到湍流的过渡颇为相似.本文指出了“过渡阶段”与“完全随机阶段”的基本区别:在“完全随机阶段”,单个情形的决定性过程对初始条件和动力学参数极端敏感,而系综的统计性质则对初始条件和动力学参数的微小变化不敏感;与此相反,在“过渡阶段”,单个情形的决定性过程和系综的统计性质对初始条件和动力学参数都敏感.造成过渡阶段这一特点的机制是在参数空间中存在着“长链结构”.本文还讨论了这一分析对其它随机现象可能具有的启示.

On the Sensitive Dynamical System and the Transition from the Apparently Deterministic Process to the Completely Random Process

The detailed analysis of the dynamical process of coin tossing is made. Through calculalions, it is illustrated how and why the result is extremely sensitive to the initial conditions. It is also shown that, as the initial height of the mass center of the coin increases, the final conflguration, i.e. "head" or "tail", becomes more and more sensitive to the initial parameters(the initial velocity angular velocity,and the initial orientation). the coefficient of the air drag, and the energy absorption factor of the surface on which the coin bounces. If we keep the "head" upward initially hut allow a small range for the change of somt other initial parameters, the frequency that thefmal conflguration is "head", would be I if the initial height h of the mass center is sufficiently small, and would be close to 1/2 if h is sufficiently large. An interesting question is how this frequency changes continuously from 1 to 1/2 as h increases. Detailed calculations show that such a "transition" is very similar to the transition from laminar to turbulent flows. A basic difference between the "transition stage" and the"completely rondom stage" is indicated: In the "completely rondom stage", the deterministic process of the individual case is extremely sensitive to the initial conditions and the dynamical parameters. but the statistical properties of the ensemble are insensitive to the small changes of the initial conditions and the dynamical parameters. On the contrary, in the "transition stage", both the deterministic process of the individual case and the statistical properties of the ensemble are sensitive to the initial conditions and the dynamical parameters. The mechanism for this feature of the "transition stage" is the existence of the "long-train structure" in the parameter space. The illuminalions of this analysis on some other rondom phenomena are discussed.