We analytically derived and confirmed by empirical data the following three relations from the quasi-time-reversal symmetry, Gibrat’s law, and the non-Gibrat’s property observed in the urban population data of France. The first is the relation between the time variation of the power law and the quasi-time-reversal symmetry in the large-scale range of a system that changes quasi-statically. The second is the relation between the time variation of the log-normal distribution and the quasi-time-reversal symmetry in the mid-scale range. The third is the relation among the parameters of log-normal distribution, non-Gibrat’s property, and quasi-time-reversal symmetry.

中文翻译：

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城市人口幂律和对数正态分布的准静态变化

我们从法国城市人口数据中观察到的准时间反转对称性、Gibrat 定律和非 Gibrat 属性，通过实证数据分析推导出并证实了以下三个关系。第一个是幂律的时间变化与准静态变化的系统的大尺度范围内的准时间反转对称性之间的关系。第二个是对数正态分布的时间变化与中尺度范围内的准时间反转对称性之间的关系。三是对数正态分布、非吉布拉特性质、准时间反演对称性等参数之间的关系。