The size distribution of cities: a kinetic explanation

TL;DR

This paper introduces a kinetic model explaining city size distributions, deriving equations that predict power law and lognormal behaviors, and validates these with data from Italy and Switzerland.

Contribution

It presents a novel kinetic approach linking migration rules to city size distributions, deriving equations that explain observed empirical laws.

Findings

Equilibrium city size distribution can follow a power law or lognormal law.

The Pareto index relates to emigration tendencies.

Model predictions align with Italian and Swiss population data.

Abstract

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.