城市位序-规模问题的分形理论初探——Zipf定律的理论来源、异化形式及其统一基础

A Fractal Study on Zipf′s Law and Rank-Size Rule of Cities in an Urban System

摘要: 从城市动力系统的一般模型出发，引出城市体系异速生长的Ｂｅｃｋｍａｎｎ方程并将其推广到一般形式，同时导出城市规模分布的Ｐａｒｅｔｏ公式，从而建立了Ｚｉｐｆ定律的理论基础；讨论了城市位序—规模问题的分形性质和退化形式，进而证明了Ｄａｖｉｓ二倍数规律与Ｍａｎｄｅｌｂｒｏｔ的三参数Ｚｉｐｆ公式等价，并修正了Ｃｕｒｒｙ最大熵模型，得到城市体系等级结构的Ｗｅｉｂｕｌ公式。

Abstract: An allometric equation of urban systems and its Beckmann’s form were deduced from the equations of general urban dynamic systems,and the Pareto′s formula of city-size distribution can be obtained out of the mathematical transformation,based on which the theoretical foundations can be laid for Zipf′s Law of urban geography.The fractal nature of rank-size rule of cities and its degenerational form were discussed,Davis′ double law (2n-law) was proved equivalent to Zipf′s formula with three parameters,and Cur...