Formation dynamics and distribution function of cities population

TL;DR

This paper analyzes city population distribution functions across different structural levels, models their dynamics using Fokker-Planck equations, and fits the results with Tsallis distribution to understand underlying stochastic processes.

Contribution

It introduces a novel model for city population dynamics based on transition rates and stochastic differential equations, linking empirical data with theoretical distributions.

Findings

Distribution functions deform with structure enlargement

Tsallis distribution fits the empirical data well

Model parameters relate to nonextensivity in population dynamics

Abstract

From the data analysis we defined distribution function against the population on the level of various structure units, namely regions, federal districts and the country on the whole. We have studied peculiarities of the distribution function deformation due to the structure units' enlargement. Using the master equation in the continuous approximation, we obtain the Fokker-Plank equation for the distribution function with symmetric transition rates. In addition, we offer a model where transitions only between neighbour states are possible. Moreover in this case it is proposed the in and out transition probability rate for any states are different We define condition for the formal equivalence of both models to the problem that is described by the stochastic differential equation with additive and multiplicative white noises. We have analyzed the corresponding Fokker-Plank equation in the Ito and Stratonovich calculus, and obtained a solution of the Tsallis distribution type. We demonstrate a comparison of theoretical results with those of the data analysis on the level a cumulative distributions. We have used the maximum likelihood method for fitting. The obtained results allow us to attribute the specific value of the Tsallis distribution nonextensivity parameter to the empiric curves.