Zipf定律、1/f噪声和分形层次
Zipf's law, 1/f noise, and fractal hierarchy
摘要：
分形、1/f噪声、齐普夫定律和大灾难事件的发生是跨各个科学的典型的普遍存在的一般经验观察，在特定科学领域内开发的一组参考文献中无法理解。所有这些观察都与标度律有关，并引起了科学界广泛的研究兴趣。然而，这些标度现象之间的内在联系仍然是有待研究的问题。本文通过理论推导和数学实验揭示了分形图案、1/f噪声和Zipf分布之间的相似性。首先，多重分形过程遵循广义齐普夫定律经验。第二，1/f频谱在数学形式上与齐普夫定律相同。第三，1/f谱和Zipf定律都可以转化为自相似的层次结构。第四，分形、1/f谱、齐普夫定律和大灾难事件的发生可以用类似的指数律和幂律来描述。自相似层次结构是一个更一般的框架或结构，可以用来包含或统一物理和社会系统中的不同标度现象和规则，如城市，河流，地震，分形，1/f噪声和秩大小分布。层次结构的数学规律可以为我们提供一个整体的视角来看待复杂性，如自组织临界性（SOC）。
英文摘要：
Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).
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