标题： 城市规模分布：一种动力学解释 显示英文标题

标题： The size distribution of cities: a kinetic explanation

摘要： 我们提出了一种关于城市集聚形成的动力学方法，该方法基于简单的迁入和迁出规则。 在大多数情况下，Boltzmann型动力学描述可以通过渐近过程得到一个具有可变扩散和漂移系数的Fokker--Planck方程，该方程描述了城市规模的概率密度随时间的演变。 结果表明，根据迁移的微观规则，平衡密度既可以遵循大尺寸变量的幂律，其中包含Zipf定律作为特例，也可以遵循中等和小尺寸变量的对数正态分布。 特别是，阐述了平衡状态下幂律的Pareto指数与人口迁出倾向之间的关系。 理论成果通过意大利和瑞士近期的人口数据进行了验证。 显示英文摘要

摘要： We present a kinetic approach to the formation of urban agglomerations which is based on simple rules of immigration and emigration. In most cases, the Boltzmann-type kinetic description allows to obtain, within an asymptotic procedure, a Fokker--Planck equation with variable coefficients of diffusion and drift, which describes the evolution in time of some probability density of the city size. It is shown that, in dependence of the microscopic rules of migration, the equilibrium density can follow both a power law for large values of the size variable, which contains as particular case a Zipf's law behavior, and a lognormal law for middle and low values of the size variable. In particular, connections between the value of Pareto index of the power law at equilibrium and the disposal of the population to emigration are outlined. The theoretical findings are tested with recent data of the populations of Italy and Switzerland.