doi:10.3390/e23070908engIshikawa A.Fujimoto S.Ramos Gutiérrez, A.Mizuno T.Quasi-static variation of power-law and log-normal distributions of urban populationART-2021-126874We 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. © 2021 by the authors.Licensee MDPI, Basel Switzerland.2021http://zaguan.unizar.es/record/11862710.3390/e23070908http://zaguan.unizar.es/record/118627oai:zaguan.unizar.es:118627ENTROPY 23, 7 (2021), 908 [16 pp]byhttp://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccess