Size distribution of cities: A kinetic explanation.

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. • We introduce a linear kinetic model for the size distribution of cities. • The elementary interactions describe the rates of immigration. • The limit of small interactions results in a Fokker–Planck equation. • The steady state can be both a tailed density and a lognormal one. • The lognormal shows a good fitting with real data from Italy and Swiss. [ABSTRACT FROM AUTHOR]