Quasi-Static Variation of Power-Law and Log-Normal Distributions of Urban Population.

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.