动力学和随机康托集的广泛分析和数值研究
Extensive analytical and numerical investigation of the kinetic and stochastic Cantor set
摘要：
我们调查，分析和数值，动力学和随机对应的三元康托集。生成器将一个区间分成三个相等的部分或随机分成三个部分并删除中间的三分之一，该生成器仅适用于一个区间，在动力学和随机康托集中的每个生成步骤中，以与其大小成比例的概率选择。我们发现，分形维数的动力学康托集与其经典的对应，尽管在空间分布的间隔有明显的差异。然而，对于随机康托集，我们发现，所得到的集的分形维数$d_f=0.56155$小于其经典值$d_f={{\ln 2}\over{\ln 3}}$。尽管如此，在所有这三种情况下，我们表明，$d_f$次幂，$d_f$是分形维数的各个集合，在所有时间的所有间隔的总和是等于一个或大小的发起人$[0,1]$，无论它是递归的，动力学或随机康托集。此外，我们提出了精确的算法，这两个变种，可以捕捉到完整的动力学所描述的速率方程用于解决各自的模型解析。我们的分析和数值模拟之间的完美协议是一个明确的证明。
英文摘要：
We investigate, both analytically and numerically, the kinetic and stochastic counterpart of the triadic Cantor set. The generator that divides an interval either into three equal pieces or into three pieces randomly and remove the middle third is applied to only one interval, picked with probability proportional to its size, at each generation step in the kinetic and stochastic Cantor set respectively. We show that the fractal dimension of the kinetic Cantor set coincides with that of its classical counterpart despite the apparent differences in the spatial distribution of the intervals. For the stochastic Cantor set, however, we find that the resulting set has fractal dimension $d_f=0.56155$ which is less than its classical value $d_f={{\ln 2}\over{\ln 3}}$. Nonetheless, in all three cases we show that the sum of the $d_f$th power, $d_f$ being the fractal dimension of the respective set, of all the intervals at all time is equal to one or the size of the initiator $[0,1]$ regardless of whether it is recursive, kinetic or stochastic Cantor set. Besides, we propose exact algorithms for both the variants which can capture the complete dynamics described by the rate equation used to solve the respective model analytically. The perfect agreement between our analytical and numerical simulation is a clear testament to that.
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