The Apollonian packings (APs) are fractals that result from a space-filling procedure with spheres. We discuss the finite size effects for finite intervals $s\in[s_\mathrm{min},s_\mathrm{max}]$ between the largest and the smallest sizes of the filling spheres. We derive a simple analytical generalization of the scale-free laws, which allows a quantitative study of such \textit{physical fractals}. To test our result, a new efficient space-filling algorithm has been developed which generates random APs of spheres with a finite range of diameters: the correct asymptotic limit $s_\mathrm{min}/s_\mathrm{max}\rightarrow 0$ and the known APs' fractal dimensions are recovered and an excellent agreement with the generalized analytic laws is proved within the overall ranges of sizes.
用动态蒙特卡罗模拟研究了扩散维数为$1.67(2)$的逾渗团簇上非重叠分形粒子的异常短时和长时自扩散.如Phys. Rev. Lett. 115，097801（2015）中，由这些颗粒形成的无序相是分形空间中的无约束、均匀和单分散流体的无序相。在热力学平衡的颗粒扩散过程中，均方化学位移随时间的非线性幂而增加，在短时间内的指数为$0.96(1)$，在长时间内的指数为$0.63(1)$。在有限的包装分数的空间位阻最近的邻居粒子之间的导致一个亚扩散制度，分离的短时间的异常扩散从长时间的异常扩散。在$\sim 60\%$和更高的填充分数的时间上观察到超过80年的颗粒定位。
The influence of packing protocol, size ratio, and pore structure on fractal exponents in dense polydisperse packings
Joint Institute for Nuclear Research（俄罗斯联邦杜布纳联合核子研究所）；Dokuchaev Soil Science Institute（多卡恰耶夫土壤科学研究所）；Horia Hulubei National Institute of Physics and Nuclear Engineering（霍里亚·胡利贝伊国家物理与核工程研究所）；Craiova University（克赖奥瓦大学）
