  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.
